# Axiomatic characterization of physical geometry by Heinz-Jurgen Schmidt

By Heinz-Jurgen Schmidt

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Laws on t h e us to e m b e d finite, Although, be (B,C,R)-level. R into disjoint it w o u l d a weakly unions distribu- of s p a t i a l regions. (2236) Proof Lemma : L e t C, A. 6 R, i = 1 . . k, C 4 B, A k + I a n d A A is the D Le£ D ~ A k + I, D (2237) Then The assertion is t r i v i a l ( k < n) C < A. k for n = I. A. 6 R. W e h a v e l > A. k+1. We conclude D > B and, that from ~ B V A k + I = A. relation V A, o n n): Definition: Therefore, C 4 A.. 1 = B V A k + I £ R b y ( 2 2 3 4 ) ( i i ) .

We (2232) A = reg(l,kA) but m { M(kl). = ~. Let ( l ' , k 1 , ~ , ~ , l " , k 2 ) def s u b s t i t u t i o n . F r o m m' A k I it f o l l o w s = pos(l",m') contradiction N o w a s s u m e A 1 = A 2. By p o s ( l , k 2) corresponding pos(l',m') (2229). say m 6 M(k2), (see = A 2. e. k A ~ k B and pos(m,nB),~,l',nB) , and n A E n B = ek B ~ ~k A we i n f e r n A = nB, A = B. : A = r e g ( l , k A) B : reg(l,kB) = reg(m,nB) C = reg(m,nc) and k A ~ As above, we c o n s i d e r kB, n B C n c. the s u b s t i t u t i o n (m,nB,pOs(l,kB),~,m',kB).

1 { m def V k 6 C(1), pos(l,k) = pos(m,k). (237) Axiom B8: Iv =C Let 1 or I v = ~ . 6 B R for v=1 . [I II. N, such that iv { iv and i I I] 12 I2 i3"''iN" JJ/~ YI / j j" 12 ///////>~P ///////// l I " Y3 r- 12 11 fig. (238) 13 50 (239) Definition: The f r a m e ~ 6 F w i l l be c a l l e d iff the f o l l o w i n g that k ~ m, property holds: F o r all k, m, In this c a s e w e w i l l w r i t e s u b s e t of i n e r t i a l pos(12,k) (k,m, ll,n) = pos(n,k). ÷ 12 . jwl/////// ,ni! m m 11 12 fig.