
theorem
  for C being non empty 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 & f1 is isomorphism
  holds p2 is isomorphism
  proof
    let C be non empty 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;
    assume
A4: f1 is isomorphism;
    consider g1 be Morphism of c,c1 such that
A5: g1 * f1 = id- c1 & f1 * g1 = id- c by A4;
    set g11 = g1 * f2;
    set g22 = id- c2;
A6: Hom(c2,c1) <> {} & Hom(c2,c2) <> {} & Hom(c1,c1) <> {} by A1,A4,Th22;
A7: f1 * g11 = (f1 * g1) * f2 by A4,A1,Th23
    .= f2 by A5,A1,Th18
    .= f2 * g22 by A1,Th18;
    then
A8: Hom(c2,d) <> {} & ex h being Morphism of c2,d st
    p1 * h = g11 & p2 * h = g22
    & for h1 being Morphism of c2,d st p1 * h1 = g11 & p2 * h1 = g22
    holds h = h1 by A2,A1,Def17,A6;
    consider q2 be Morphism of c2,d such that
A9: p1 * q2 = g11 & p2 * q2 = g22
    & for h1 being Morphism of c2,d st p1 * h1 = g11 & p2 * h1 = g22
    holds q2 = h1 by A6,A2,A1,Def17,A7;
    set g33 = p1 * q2 * p2;
A10: Hom(d,c) <> {} by A1,Th22;
    f1 * g33 = f1 * (g1 * (f2 * p2)) by A9,A4,A1,Th23
    .= (f1 * g1) * (f2 * p2) by A10,A4,Th23
    .= f2 * p2 by A10,Th18,A5;
    then consider h be Morphism of d,d such that
    p1 * h = g33 & p2 * h = p2 and
A11: for h1 being Morphism of d,d st p1 * h1 = g33 & p2 * h1 = p2 holds h = h1
     by A1,A2,Def17;
A12: p1 * id- d = p1 by A1,Th18
    .= g1 * f1 * p1 by A1,A5,Th18
    .= g1 * (f1 * p1) by A1,A4,Th23
    .= g33 by A9,A3,A4,A1,Th23;
A13: p2 * id- d = p2 by A1,Th18;
A14: p1 * (q2 * p2) = g33 by A1,A8,Th23;
    p2 * (q2 * p2) = p2 * q2 * p2 by A1,A8,Th23
    .= p2 by A1,A9,Th18;
    then h = q2 * p2 by A11,A14;
    hence p2 is isomorphism by A7,A9,A13,A11,A12,A2,A1,Def17,A6;
  end;
