
theorem
  for C being category, c,c1,c2,d being Object of C,
      f1 being Morphism of c1,c, f2 being Morphism of c2,c,
      p1 being Morphism of d,c1, 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
  proof
    let C be category;
    let c,c1,c2,d be Object of C;
    let f1 be Morphism of c1,c;
    let f2 be Morphism of c2,c;
    let p1 be Morphism of d,c1;
    let p2 be Morphism of d,c2;
    assume
A1: Hom(c1,c) <> {} & Hom(c2,c) <> {} & Hom(d,c1) <> {} & Hom(d,c2) <> {};
    assume
A2: d,p1,p2 is_pullback_of f1,f2;
    then
A3: f1 * p1 = f2 * p2 & for d1 being Object of C,
    g1 being Morphism of d1,c1, 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,
    g2 being Morphism of d1,c2, 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;
      let g2 be Morphism of d1,c2;
      let g1 be Morphism of d1,c1;
      assume
A4:   Hom(d1,c2) <> {} & Hom(d1,c1) <> {} & f2 * g2 = f1 * g1;
      hence Hom(d1,d) <> {} by A2,A1,Def17;
      consider h be 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;
      thus thesis by A5;
    end;
    hence d,p2,p1 is_pullback_of f2,f1 by A3,A1,Def17;
  end;
