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_2:15
.= b by Th2 ;
A6: g " is dilatation by A2, Th70;
assume A7: a <> b ; :: thesis: f = g
((g ") * f) . a = (g ") . (g . a) by A3, FUNCT_2:15
.= a by Th2 ;
then A8: (g ") * f = id the carrier of AFS by A1, A7, A5, A6, Th71, Th78;

now :: thesis: for x being Element of AFS holds g . x = f . x

let x be Element of AFS; :: thesis: g . x = f . x
(g ") . (f . x) = ((g ") * f) . x by FUNCT_2:15;
then (g ") . (f . x) = x by A8;
hence g . x = f . x by Th2; :: thesis: verum

end;

hence f = g by FUNCT_2:63; :: thesis: verum