let A be non empty set ; :: thesis: for f, g being Permutation of A
for R being Relation of [:A,A:] st ( for a, b, x, y, z, t being Element of A st [[x,y],[a,b]] in R & [[a,b],[z,t]] in R & a <> b holds
[[x,y],[z,t]] in R ) & ( for x, y, z being Element of A holds [[x,x],[y,z]] in R ) & f is_FormalIz_of R & g is_FormalIz_of R holds
f * g is_FormalIz_of R

let f, g be Permutation of A; :: thesis: for R being Relation of [:A,A:] st ( for a, b, x, y, z, t being Element of A st [[x,y],[a,b]] in R & [[a,b],[z,t]] in R & a <> b holds
[[x,y],[z,t]] in R ) & ( for x, y, z being Element of A holds [[x,x],[y,z]] in R ) & f is_FormalIz_of R & g is_FormalIz_of R holds
f * g is_FormalIz_of R

let R be Relation of [:A,A:]; :: thesis: ( ( for a, b, x, y, z, t being Element of A st [[x,y],[a,b]] in R & [[a,b],[z,t]] in R & a <> b holds
[[x,y],[z,t]] in R ) & ( for x, y, z being Element of A holds [[x,x],[y,z]] in R ) & f is_FormalIz_of R & g is_FormalIz_of R implies f * g is_FormalIz_of R )

assume that
A1: for a, b, x, y, z, t being Element of A st [[x,y],[a,b]] in R & [[a,b],[z,t]] in R & a <> b holds
[[x,y],[z,t]] in R and
A2: for x, y, z being Element of A holds [[x,x],[y,z]] in R and
A3: for x, y being Element of A holds [[x,y],[(f . x),(f . y)]] in R and
A4: for x, y being Element of A holds [[x,y],[(g . x),(g . y)]] in R ; :: according to TRANSGEO:def 2 :: thesis:
let x be Element of A; :: according to TRANSGEO:def 2 :: thesis: for y being Element of A holds [[x,y],[((f * g) . x),((f * g) . y)]] in R
let y be Element of A; :: thesis: [[x,y],[((f * g) . x),((f * g) . y)]] in R
( f . (g . x) = (f * g) . x & f . (g . y) = (f * g) . y ) by FUNCT_2:15;
then A5: [[(g . x),(g . y)],[((f * g) . x),((f * g) . y)]] in R by A3;
A6: now :: thesis: ( g . x = g . y implies [[x,y],[((f * g) . x),((f * g) . y)]] in R )
assume g . x = g . y ; :: thesis: [[x,y],[((f * g) . x),((f * g) . y)]] in R
then x = y by FUNCT_2:58;
hence [[x,y],[((f * g) . x),((f * g) . y)]] in R by A2; :: thesis: verum
end;
[[x,y],[(g . x),(g . y)]] in R by A4;
hence [[x,y],[((f * g) . x),((f * g) . y)]] in R by A1, A5, A6; :: thesis: verum