let AFS be AffinSpace; :: thesis: for a, x being Element of AFS

for f, g being Permutation of the carrier of AFS st f is translation & g is translation & f . a = g . a & not LIN a,f . a,x holds

f . x = g . x

let a, x be Element of AFS; :: thesis: for f, g being Permutation of the carrier of AFS st f is translation & g is translation & f . a = g . a & not LIN a,f . a,x holds

f . x = g . x

let f, g be Permutation of the carrier of AFS; :: thesis: ( f is translation & g is translation & f . a = g . a & not LIN a,f . a,x implies f . x = g . x )

assume that

A1: f is translation and

A2: g is translation and

A3: f . a = g . a and

A4: not LIN a,f . a,x ; :: thesis: f . x = g . x

set b = f . a;

set y = f . x;

set z = g . x;

A5: ( a,x // f . a,g . x & a,f . a // x,g . x ) by A2, A3, Th68, Th82;

f is dilatation by A1;

then A6: a,x // f . a,f . x by Th68;

a,f . a // x,f . x by A1, Th82;

hence f . x = g . x by A4, A6, A5, Th80; :: thesis: verum

for f, g being Permutation of the carrier of AFS st f is translation & g is translation & f . a = g . a & not LIN a,f . a,x holds

f . x = g . x

let a, x be Element of AFS; :: thesis: for f, g being Permutation of the carrier of AFS st f is translation & g is translation & f . a = g . a & not LIN a,f . a,x holds

f . x = g . x

let f, g be Permutation of the carrier of AFS; :: thesis: ( f is translation & g is translation & f . a = g . a & not LIN a,f . a,x implies f . x = g . x )

assume that

A1: f is translation and

A2: g is translation and

A3: f . a = g . a and

A4: not LIN a,f . a,x ; :: thesis: f . x = g . x

set b = f . a;

set y = f . x;

set z = g . x;

A5: ( a,x // f . a,g . x & a,f . a // x,g . x ) by A2, A3, Th68, Th82;

f is dilatation by A1;

then A6: a,x // f . a,f . x by Th68;

a,f . a // x,f . x by A1, Th82;

hence f . x = g . x by A4, A6, A5, Th80; :: thesis: verum