MATHEMATICAL CONCEPTS OF THE MATERIAL WOELD. 499 



*10'4. TT Hp (j> is the statement that, if u is a subclass ofO^,, and v is <f>-axial and 

 contained in cm^'u, then there exists a class w (possibly the null class) such that the 

 logical sum ofv and w is (j>-axial and ^-equivalent to u. In symbols, 



it Hp <fr . = : u e cls'O . v e ax,,, n cls'cm^'tt .=,. (g;w) . v u weax^, n equiv^'w Df 



*10'5. />Hp< is the statement that ifu and v are both (f>-axial, and if they possess 

 at least two members in common, then their logical sum is <f>-maximal, In symbols, 



p Tip <ft . = : u, v e ax . NC'(M n v) = 2 . D Bit) . u u v e mx Df 



Elucidatory Note. Referring to the previous elucidatory notes (where < is flatness), 

 we see that *10'4 in effect assumes that a line can always be added (1) to two con- 

 current lines to form a set of three concurrent non-coplanar lines, and (2) to one line in 

 a plane to form a set of two concurrent lines in that plane. Also *10'5 assumes that, 

 if two sets of three concurrent lines have two members in common, the four lines are 

 concurrent. 



Deductions from the Axioms. 

 *11. Preliminary Propositions. 



*11'11. Proposition.- Assuming (X, i>)Hp<, O^, has at least three members. In 

 symbols, 



I- : (X, v) Hp (/>.=>. Nc'O^ 3 



Proof Of. *4-27 and *10'1'3. 



*11'12. Proposition. Assuming (X, /i, v)Hp<f), cm^'A is the null class (A). In 

 symbols, 



|- : (X, JLI, v) Up <.=>. cm/A = A 



Proof Cf *4'28 and *10'2 and *1M1. 



*1T21. Proposition. Assuming (X, /i) Hp <f>, all ^-prime classes with more tlian one 

 member are contained in O^,. In symbols, 



I" . '. (X, p.) Hp <f> . D : v e prm^, . Nc't' > 1 . ID . v e cls'O^, 



Proof. For if a; is not a member of O^,, then cm^Ya; is the class of all entities. 

 Hence (cf. *4'21'27 and *10'1'2) the conclusion follows. 



*12. On $- Axial Classes and $- Dimensions. 



*12'11. Proposition. Assuming (X, p., v) Hp <j>, every unit class whose single 

 member belongs to O^, is ^-axial. In symbols, 



|~ . '. (X, /A, v) Hp (j> . => : x e O^, . D T . L'X e ax^ 



Proof. Cf. *3'21-22 and *4'27 and *5'233 and *10'2. 



*12'12. Proposition. Assuming (X-Tr)Hp^, every subclass of O^,, not the null 



3 s 2 



