let OAS be OAffinSpace; :: thesis: for f being Permutation of the carrier of OAS st f is dilatation holds
( ( f = id the carrier of OAS or for x being Element of OAS holds f . x <> x ) iff for x, y being Element of OAS holds x,f . x '||' y,f . y )
let f be Permutation of the carrier of OAS; :: thesis: ( f is dilatation implies ( ( f = id the carrier of OAS or for x being Element of OAS holds f . x <> x ) iff for x, y being Element of OAS holds x,f . x '||' y,f . y ) )
assume A1: f is dilatation ; :: thesis: ( ( f = id the carrier of OAS or for x being Element of OAS holds f . x <> x ) iff for x, y being Element of OAS holds x,f . x '||' y,f . y )

A2: now

assume A3: for x, y being Element of OAS holds x,f . x '||' y,f . y ; :: thesis: ( f <> id the carrier of OAS implies for x being Element of OAS holds not f . x = x )
assume f <> id the carrier of OAS ; :: thesis: for x being Element of OAS holds not f . x = x
then consider y being Element of OAS such that
A4: f . y <> (id the carrier of OAS) . y by FUNCT_2:63;
given x being Element of OAS such that A5: f . x = x ; :: thesis: contradiction
x <> y by A5, A4, FUNCT_1:18;
then consider z being Element of OAS such that
A6: not LIN x,y,z by DIRAF:37;
x,z '||' f . x,f . z by A1, Th51;
then LIN x,z,f . z by A5, DIRAF:def 5;
then A7: LIN z,f . z,x by DIRAF:30;
LIN z,f . z,z by DIRAF:31;
then A8: z,f . z '||' x,z by A7, DIRAF:34;
A9: f . y <> y by A4, FUNCT_1:18;
x,y '||' f . x,f . y by A1, Th51;
then A10: LIN x,y,f . y by A5, DIRAF:def 5;
then LIN y,x,f . y by DIRAF:30;
then A11: y,x '||' y,f . y by DIRAF:def 5;
A12: LIN y,f . y,x by A10, DIRAF:30;

A13: now

assume z = f . z ; :: thesis: contradiction
then z,y '||' z,f . y by A1, Th51;
then LIN z,y,f . y by DIRAF:def 5;
then ( LIN y,f . y,y & LIN y,f . y,z ) by DIRAF:30, DIRAF:31;
hence contradiction by A9, A12, A6, DIRAF:32; :: thesis: verum

end;

y,f . y '||' z,f . z by A3;
then y,f . y '||' x,z by A13, A8, DIRAF:23;
then y,x '||' x,z by A9, A11, DIRAF:23;
then x,y '||' x,z by DIRAF:22;
hence contradiction by A6, DIRAF:def 5; :: thesis: verum

end;

now

assume A14: ( f = id the carrier of OAS or for z being Element of OAS holds f . z <> z ) ; :: thesis: for x, y being Element of OAS holds x,f . x '||' y,f . y
let x, y be Element of OAS; :: thesis: x,f . x '||' y,f . y
A15: x,y '||' f . x,f . y by A1, Th51;

A16: now

assume A17: for z being Element of OAS 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 OAS such that
A19: LIN x,f . x,z and
A20: LIN y,f . y,z by A15, Th61;
set t = f . z;
LIN x,f . x,f . z by A1, A19, Th60;
then A21: x,f . x '||' z,f . z by A19, DIRAF:34;
LIN y,f . y,f . z by A1, A20, Th60;
then A22: y,f . y '||' z,f . z by A20, DIRAF:34;
z <> f . z by A17;
hence contradiction by A18, A21, A22, DIRAF:23; :: thesis: verum

end;

now

assume f = id the carrier of OAS ; :: thesis: x,f . x '||' y,f . y
then f . y = y by FUNCT_1:18;
hence x,f . x '||' y,f . y by DIRAF:20; :: thesis: verum

end;

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

end;

hence ( ( f = id the carrier of OAS or for x being Element of OAS holds f . x <> x ) iff for x, y being Element of OAS holds x,f . x '||' y,f . y ) by A2; :: thesis: verum