let C be category; :: thesis: for c, c1, c2, d being Object of C
for f1 being Morphism of c1,c
for f2 being Morphism of c2,c
for p1 being Morphism of d,c1
for p2 being Morphism of d,c2 st Hom (c1,c) <> {} & Hom (c2,c) <> {} & Hom (d,c1) <> {} & Hom (d,c2) <> {} & d,p1,p2 is_pullback_of f1,f2 holds
d,p2,p1 is_pullback_of f2,f1
let c, c1, c2, d be Object of C; :: thesis: for f1 being Morphism of c1,c
for f2 being Morphism of c2,c
for p1 being Morphism of d,c1
for p2 being Morphism of d,c2 st Hom (c1,c) <> {} & Hom (c2,c) <> {} & Hom (d,c1) <> {} & Hom (d,c2) <> {} & d,p1,p2 is_pullback_of f1,f2 holds
d,p2,p1 is_pullback_of f2,f1
let f1 be Morphism of c1,c; :: thesis: for f2 being Morphism of c2,c
for p1 being Morphism of d,c1
for p2 being Morphism of d,c2 st Hom (c1,c) <> {} & Hom (c2,c) <> {} & Hom (d,c1) <> {} & Hom (d,c2) <> {} & d,p1,p2 is_pullback_of f1,f2 holds
d,p2,p1 is_pullback_of f2,f1
let f2 be Morphism of c2,c; :: thesis: for p1 being Morphism of d,c1
for p2 being Morphism of d,c2 st Hom (c1,c) <> {} & Hom (c2,c) <> {} & Hom (d,c1) <> {} & Hom (d,c2) <> {} & d,p1,p2 is_pullback_of f1,f2 holds
d,p2,p1 is_pullback_of f2,f1
let p1 be Morphism of d,c1; :: thesis: for p2 being Morphism of d,c2 st Hom (c1,c) <> {} & Hom (c2,c) <> {} & Hom (d,c1) <> {} & Hom (d,c2) <> {} & d,p1,p2 is_pullback_of f1,f2 holds
d,p2,p1 is_pullback_of f2,f1
let p2 be Morphism of d,c2; :: thesis: ( Hom (c1,c) <> {} & Hom (c2,c) <> {} & Hom (d,c1) <> {} & Hom (d,c2) <> {} & d,p1,p2 is_pullback_of f1,f2 implies d,p2,p1 is_pullback_of f2,f1 )
assume A1: ( Hom (c1,c) <> {} & Hom (c2,c) <> {} & Hom (d,c1) <> {} & Hom (d,c2) <> {} ) ; :: thesis: ( not d,p1,p2 is_pullback_of f1,f2 or d,p2,p1 is_pullback_of f2,f1 )
assume A2: d,p1,p2 is_pullback_of f1,f2 ; :: thesis: d,p2,p1 is_pullback_of f2,f1
then A3: ( f1 * p1 = f2 * p2 & ( for d1 being Object of C
for g1 being Morphism of d1,c1
for g2 being Morphism of d1,c2 st Hom (d1,c1) <> {} & Hom (d1,c2) <> {} & f1 * g1 = f2 * g2 holds
( Hom (d1,d) <> {} & ex h being Morphism of d1,d st
( p1 * h = g1 & p2 * h = g2 & ( for h1 being Morphism of d1,d st p1 * h1 = g1 & p2 * h1 = g2 holds
h = h1 ) ) ) ) ) by A1, Def17;
for d1 being Object of C
for g2 being Morphism of d1,c2
for g1 being Morphism of d1,c1 st Hom (d1,c2) <> {} & Hom (d1,c1) <> {} & f2 * g2 = f1 * g1 holds
( Hom (d1,d) <> {} & ex h being Morphism of d1,d st
( p2 * h = g2 & p1 * h = g1 & ( for h1 being Morphism of d1,d st p2 * h1 = g2 & p1 * h1 = g1 holds
h = h1 ) ) )

proof

let d1 be Object of C; :: thesis: for g2 being Morphism of d1,c2
for g1 being Morphism of d1,c1 st Hom (d1,c2) <> {} & Hom (d1,c1) <> {} & f2 * g2 = f1 * g1 holds
( Hom (d1,d) <> {} & ex h being Morphism of d1,d st
( p2 * h = g2 & p1 * h = g1 & ( for h1 being Morphism of d1,d st p2 * h1 = g2 & p1 * h1 = g1 holds
h = h1 ) ) )
let g2 be Morphism of d1,c2; :: thesis: for g1 being Morphism of d1,c1 st Hom (d1,c2) <> {} & Hom (d1,c1) <> {} & f2 * g2 = f1 * g1 holds
( Hom (d1,d) <> {} & ex h being Morphism of d1,d st
( p2 * h = g2 & p1 * h = g1 & ( for h1 being Morphism of d1,d st p2 * h1 = g2 & p1 * h1 = g1 holds
h = h1 ) ) )
let g1 be Morphism of d1,c1; :: thesis: ( Hom (d1,c2) <> {} & Hom (d1,c1) <> {} & f2 * g2 = f1 * g1 implies ( Hom (d1,d) <> {} & ex h being Morphism of d1,d st
( p2 * h = g2 & p1 * h = g1 & ( for h1 being Morphism of d1,d st p2 * h1 = g2 & p1 * h1 = g1 holds
h = h1 ) ) ) )
assume A4: ( Hom (d1,c2) <> {} & Hom (d1,c1) <> {} & f2 * g2 = f1 * g1 ) ; :: thesis: ( Hom (d1,d) <> {} & ex h being Morphism of d1,d st
( p2 * h = g2 & p1 * h = g1 & ( for h1 being Morphism of d1,d st p2 * h1 = g2 & p1 * h1 = g1 holds
h = h1 ) ) )
hence Hom (d1,d) <> {} by A2, A1, Def17; :: thesis: ex h being Morphism of d1,d st
( p2 * h = g2 & p1 * h = g1 & ( for h1 being Morphism of d1,d st p2 * h1 = g2 & p1 * h1 = g1 holds
h = h1 ) )
consider h being Morphism of d1,d such that
A5: ( p1 * h = g1 & p2 * h = g2 & ( for h1 being Morphism of d1,d st p1 * h1 = g1 & p2 * h1 = g2 holds
h = h1 ) ) by A4, A2, A1, Def17;
take h ; :: thesis: ( p2 * h = g2 & p1 * h = g1 & ( for h1 being Morphism of d1,d st p2 * h1 = g2 & p1 * h1 = g1 holds
h = h1 ) )
thus ( p2 * h = g2 & p1 * h = g1 & ( for h1 being Morphism of d1,d st p2 * h1 = g2 & p1 * h1 = g1 holds
h = h1 ) ) by A5; :: thesis: verum

end;

hence d,p2,p1 is_pullback_of f2,f1 by A3, A1, Def17; :: thesis: verum