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: f * g is_FormalIz_of R

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;

hence [[x,y],[((f * g) . x),((f * g) . y)]] in R by A1, A5, A6; :: thesis: verum

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: f * g is_FormalIz_of R

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 )

[[x,y],[(g . x),(g . y)]] in R
by A4;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;then x = y by FUNCT_2:58;

hence [[x,y],[((f * g) . x),((f * g) . y)]] in R by A2; :: thesis: verum

hence [[x,y],[((f * g) . x),((f * g) . y)]] in R by A1, A5, A6; :: thesis: verum