let AFS be AffinSpace; :: thesis: for f being Permutation of the carrier of AFS st f is dilatation holds

( ( f = id the carrier of AFS or for x being Element of AFS holds f . x <> x ) iff for x, y being Element of AFS holds x,f . x // y,f . y )

let f be Permutation of the carrier of AFS; :: thesis: ( f is dilatation implies ( ( f = id the carrier of AFS or for x being Element of AFS holds f . x <> x ) iff for x, y being Element of AFS holds x,f . x // y,f . y ) )

assume A1: f is dilatation ; :: thesis: ( ( f = id the carrier of AFS or for x being Element of AFS holds f . x <> x ) iff for x, y being Element of AFS holds x,f . x // y,f . y )

( ( f = id the carrier of AFS or for x being Element of AFS holds f . x <> x ) iff for x, y being Element of AFS holds x,f . x // y,f . y )

let f be Permutation of the carrier of AFS; :: thesis: ( f is dilatation implies ( ( f = id the carrier of AFS or for x being Element of AFS holds f . x <> x ) iff for x, y being Element of AFS holds x,f . x // y,f . y ) )

assume A1: f is dilatation ; :: thesis: ( ( f = id the carrier of AFS or for x being Element of AFS holds f . x <> x ) iff for x, y being Element of AFS holds x,f . x // y,f . y )

A2: now :: thesis: ( ( for x, y being Element of AFS holds x,f . x // y,f . y ) & f <> id the carrier of AFS implies for x being Element of AFS holds not f . x = x )

assume A3:
for x, y being Element of AFS holds x,f . x // y,f . y
; :: thesis: ( f <> id the carrier of AFS implies for x being Element of AFS holds not f . x = x )

assume f <> id the carrier of AFS ; :: thesis: for x being Element of AFS holds not f . x = x

then consider y being Element of AFS such that

A4: f . y <> (id the carrier of AFS) . y by FUNCT_2:63;

given x being Element of AFS such that A5: f . x = x ; :: thesis: contradiction

x <> y by A5, A4;

then consider z being Element of AFS such that

A6: not LIN x,y,z by AFF_1:13;

x,z // f . x,f . z by A1, Th68;

then LIN x,z,f . z by A5, AFF_1:def 1;

then A7: LIN z,f . z,x by AFF_1:6;

LIN z,f . z,z by AFF_1:7;

then A8: z,f . z // x,z by A7, AFF_1:10;

A9: f . y <> y by A4;

x,y // f . x,f . y by A1, Th68;

then A10: LIN x,y,f . y by A5, AFF_1:def 1;

then LIN y,x,f . y by AFF_1:6;

then A11: y,x // y,f . y by AFF_1:def 1;

A12: LIN y,f . y,x by A10, AFF_1:6;

then y,f . y // x,z by A13, A8, AFF_1:5;

then y,x // x,z by A9, A11, AFF_1:5;

then x,y // x,z by AFF_1:4;

hence contradiction by A6, AFF_1:def 1; :: thesis: verum

end;assume f <> id the carrier of AFS ; :: thesis: for x being Element of AFS holds not f . x = x

then consider y being Element of AFS such that

A4: f . y <> (id the carrier of AFS) . y by FUNCT_2:63;

given x being Element of AFS such that A5: f . x = x ; :: thesis: contradiction

x <> y by A5, A4;

then consider z being Element of AFS such that

A6: not LIN x,y,z by AFF_1:13;

x,z // f . x,f . z by A1, Th68;

then LIN x,z,f . z by A5, AFF_1:def 1;

then A7: LIN z,f . z,x by AFF_1:6;

LIN z,f . z,z by AFF_1:7;

then A8: z,f . z // x,z by A7, AFF_1:10;

A9: f . y <> y by A4;

x,y // f . x,f . y by A1, Th68;

then A10: LIN x,y,f . y by A5, AFF_1:def 1;

then LIN y,x,f . y by AFF_1:6;

then A11: y,x // y,f . y by AFF_1:def 1;

A12: LIN y,f . y,x by A10, AFF_1:6;

A13: now :: thesis: not z = f . z

y,f . y // z,f . z
by A3;assume
z = f . z
; :: thesis: contradiction

then z,y // z,f . y by A1, Th68;

then LIN z,y,f . y by AFF_1:def 1;

then ( LIN y,f . y,y & LIN y,f . y,z ) by AFF_1:6, AFF_1:7;

hence contradiction by A9, A12, A6, AFF_1:8; :: thesis: verum

end;then z,y // z,f . y by A1, Th68;

then LIN z,y,f . y by AFF_1:def 1;

then ( LIN y,f . y,y & LIN y,f . y,z ) by AFF_1:6, AFF_1:7;

hence contradiction by A9, A12, A6, AFF_1:8; :: thesis: verum

then y,f . y // x,z by A13, A8, AFF_1:5;

then y,x // x,z by A9, A11, AFF_1:5;

then x,y // x,z by AFF_1:4;

hence contradiction by A6, AFF_1:def 1; :: thesis: verum

now :: thesis: ( ( f = id the carrier of AFS or for z being Element of AFS holds f . z <> z ) implies for x, y being Element of AFS holds x,f . x // y,f . y )

hence
( ( f = id the carrier of AFS or for x being Element of AFS holds f . x <> x ) iff for x, y being Element of AFS holds x,f . x // y,f . y )
by A2; :: thesis: verumassume A14:
( f = id the carrier of AFS or for z being Element of AFS holds f . z <> z )
; :: thesis: for x, y being Element of AFS holds x,f . x // y,f . y

let x, y be Element of AFS; :: thesis: x,f . x // y,f . y

A15: x,y // f . x,f . y by A1, Th68;

hence x,f . x // y,f . y by A14, A16; :: thesis: verum

end;let x, y be Element of AFS; :: thesis: x,f . x // y,f . y

A15: x,y // f . x,f . y by A1, Th68;

A16: now :: thesis: ( ( for z being Element of AFS holds f . z <> z ) implies x,f . x // y,f . y )

( f = id the carrier of AFS implies x,f . x // y,f . y )
by AFF_1:3;assume A17:
for z being Element of AFS holds f . z <> z
; :: thesis: x,f . x // y,f . y

assume A18: not x,f . x // y,f . y ; :: thesis: contradiction

then consider z being Element of AFS such that

A19: LIN x,f . x,z and

A20: LIN y,f . y,z by A15, Th75;

set t = f . z;

LIN x,f . x,f . z by A1, A19, Th74;

then A21: x,f . x // z,f . z by A19, AFF_1:10;

LIN y,f . y,f . z by A1, A20, Th74;

then A22: y,f . y // z,f . z by A20, AFF_1:10;

z <> f . z by A17;

hence contradiction by A18, A21, A22, AFF_1:5; :: thesis: verum

end;assume A18: not x,f . x // y,f . y ; :: thesis: contradiction

then consider z being Element of AFS such that

A19: LIN x,f . x,z and

A20: LIN y,f . y,z by A15, Th75;

set t = f . z;

LIN x,f . x,f . z by A1, A19, Th74;

then A21: x,f . x // z,f . z by A19, AFF_1:10;

LIN y,f . y,f . z by A1, A20, Th74;

then A22: y,f . y // z,f . z by A20, AFF_1:10;

z <> f . z by A17;

hence contradiction by A18, A21, A22, AFF_1:5; :: thesis: verum

hence x,f . x // y,f . y by A14, A16; :: thesis: verum