let AFS be AffinSpace; :: thesis: for a, b being Element of AFS
for f, g being Permutation of the carrier of AFS st f is dilatation & g is dilatation & f . a = g . a & f . b = g . b & not a = b holds
f = g
let a, b be Element of AFS; :: thesis: for f, g being Permutation of the carrier of AFS st f is dilatation & g is dilatation & f . a = g . a & f . b = g . b & not a = b holds
f = g
let f, g be Permutation of the carrier of AFS; :: thesis: ( f is dilatation & g is dilatation & f . a = g . a & f . b = g . b & not a = b implies f = g )
assume that
A1: f is dilatation and
A2: g is dilatation and
A3: f . a = g . a and
A4: f . b = g . b ; :: thesis: ( a = b or f = g )
A5: ((g " ) * f) . b = (g " ) . (g . b) by A4, FUNCT_2:21
.= b by Th4 ;
A6: g " is dilatation by A2, Th87;
assume A7: a <> b ; :: thesis: f = g
((g " ) * f) . a = (g " ) . (g . a) by A3, FUNCT_2:21
.= a by Th4 ;
then A8: (g " ) * f = id the carrier of AFS by A1, A7, A5, A6, Th88, Th95;

now

let x be Element of AFS; :: thesis: g . x = f . x
(g " ) . (f . x) = ((g " ) * f) . x by FUNCT_2:21;
then (g " ) . (f . x) = x by A8, FUNCT_1:35;
hence g . x = f . x by Th4; :: thesis: verum

end;

hence f = g by FUNCT_2:113; :: thesis: verum