let A be non empty set ; :: thesis: for f being Permutation of A
for R being Relation of [:A,A:] st R is_symmetric_in [:A,A:] & R is_transitive_in [:A,A:] & f is_FormalIz_of R holds
f is_automorphism_of R
let f be Permutation of A; :: thesis: for R being Relation of [:A,A:] st R is_symmetric_in [:A,A:] & R is_transitive_in [:A,A:] & f is_FormalIz_of R holds
f is_automorphism_of R
let R be Relation of [:A,A:]; :: thesis: ( R is_symmetric_in [:A,A:] & R is_transitive_in [:A,A:] & f is_FormalIz_of R implies f is_automorphism_of R )
assume that
A1: for x, y being set st x in [:A,A:] & y in [:A,A:] & [x,y] in R holds
[y,x] in R and
A2: for x, y, z being set st x in [:A,A:] & y in [:A,A:] & z in [:A,A:] & [x,y] in R & [y,z] in R holds
[x,z] in R and
A3: for x, y being Element of A holds [[x,y],[(f . x),(f . y)]] in R ; :: according to RELAT_2:def 3,RELAT_2:def 8,TRANSGEO:def 2 :: thesis: f is_automorphism_of R
let x be Element of A; :: according to TRANSGEO:def 3 :: thesis: for y, z, t being Element of A holds
( [[x,y],[z,t]] in R iff [[(f . x),(f . y)],[(f . z),(f . t)]] in R )
let y, z, t be Element of A; :: thesis: ( [[x,y],[z,t]] in R iff [[(f . x),(f . y)],[(f . z),(f . t)]] in R )
A4: ( [x,y] in [:A,A:] & [z,t] in [:A,A:] & [(f . x),(f . y)] in [:A,A:] & [(f . z),(f . t)] in [:A,A:] ) by ZFMISC_1:def 2;

A5: now

assume A6: [[x,y],[z,t]] in R ; :: thesis: [[(f . x),(f . y)],[(f . z),(f . t)]] in R
( [[x,y],[(f . x),(f . y)]] in R & [[z,t],[(f . z),(f . t)]] in R ) by A3;
then ( [[(f . x),(f . y)],[x,y]] in R & [[z,t],[(f . z),(f . t)]] in R ) by A1, A4;
then ( [[(f . x),(f . y)],[z,t]] in R & [[z,t],[(f . z),(f . t)]] in R ) by A2, A4, A6;
hence [[(f . x),(f . y)],[(f . z),(f . t)]] in R by A2, A4; :: thesis: verum

end;

now

assume A7: [[(f . x),(f . y)],[(f . z),(f . t)]] in R ; :: thesis: [[x,y],[z,t]] in R
( [[x,y],[(f . x),(f . y)]] in R & [[z,t],[(f . z),(f . t)]] in R ) by A3;
then ( [[x,y],[(f . z),(f . t)]] in R & [[(f . z),(f . t)],[z,t]] in R ) by A1, A2, A4, A7;
hence [[x,y],[z,t]] in R by A2, A4; :: thesis: verum

end;

hence ( [[x,y],[z,t]] in R iff [[(f . x),(f . y)],[(f . z),(f . t)]] in R ) by A5; :: thesis: verum