
theorem
  for C1,C2,A,B being category, P1 be Functor of A,C1, P2 be Functor of A,C2,
      Q1 be Functor of B,C1, Q2 be Functor of B,C2
  st P1 is covariant & P2 is covariant & Q1 is covariant & Q2 is covariant &
      A,P1,P2 is_product_of C1,C2 & B,Q1,Q2 is_product_of C1,C2
  holds A ~= B
  proof
    let C1,C2,A,B be category;
    let P1 be Functor of A,C1;
    let P2 be Functor of A,C2;
    let Q1 be Functor of B,C1;
    let Q2 be Functor of B,C2;
    assume
A1: P1 is covariant & P2 is covariant & Q1 is covariant & Q2 is covariant;
    assume
A2: A,P1,P2 is_product_of C1,C2;
    assume
A3: B,Q1,Q2 is_product_of C1,C2;
    ex FF being Functor of A,B, GG being Functor of B,A st
    FF is covariant & GG is covariant & GG (*) FF = id A & FF (*) GG = id B
    proof
      consider FF be Functor of A,B such that
A4:   FF is covariant & Q1 (*) FF = P1 & Q2 (*) FF = P2 &
      for H1 being Functor of A,B st H1 is covariant &
      Q1 (*) H1 = P1 & Q2 (*) H1 = P2 holds FF = H1 by A1,A3,Def17;
      consider GG be Functor of B,A such that
A5:   GG is covariant & P1 (*) GG = Q1 & P2 (*) GG = Q2 &
      for H1 being Functor of B,A st H1 is covariant &
      P1 (*) H1 = Q1 & P2 (*) H1 = Q2 holds GG = H1 by A1,A2,Def17;
      take FF,GG;
      thus FF is covariant & GG is covariant by A4,A5;
      set G11 = Q1 (*) FF;
      set G12 = Q2 (*) FF;
      consider H1 be Functor of A,A such that
A6:   H1 is covariant & P1 (*) H1 = G11 & P2 (*) H1 = G12 &
      for H being Functor of A,A st H is covariant & P1 (*) H = G11 &
      P2 (*) H = G12 holds H1 = H by A1,A4,A2,Def17;
A7:   P1 (*) (GG (*) FF) = G11 by A1,A4,A5,CAT_7:10;
A8:  P2 (*) (GG (*) FF) = G12 by A1,A4,A5,CAT_7:10;
A9:  P1 (*) id A = G11 by A1,A4,CAT_7:11;
A10:  P2 (*) id A = G12 by A1,A4,CAT_7:11;
      thus GG (*) FF = H1 by A6,A7,A8,A4,A5,CAT_6:35 .= id A by A6,A9,A10;
      set G21 = P1 (*) GG;
      set G22 = P2 (*) GG;
      consider H2 be Functor of B,B such that
A11:   H2 is covariant & Q1 (*) H2 = G21 & Q2 (*) H2 = G22 &
      for H being Functor of B,B st H is covariant & Q1 (*) H = G21 &
      Q2 (*) H = G22 holds H2 = H by A1,A5,A3,Def17;
A12:  Q1 (*) (FF (*) GG) = G21 by A1,A4,A5,CAT_7:10;
A13:  Q2 (*) (FF (*) GG) = G22 by A1,A4,A5,CAT_7:10;
A14:  Q1 (*) id B = G21 by A1,A5,CAT_7:11;
A15:  Q2 (*) id B = G22 by A1,A5,CAT_7:11;
      thus FF (*) GG = H2 by A11,A12,A13,A4,A5,CAT_6:35
      .= id B by A11,A14,A15;
    end;
    hence A ~= B by CAT_6:def 28;
  end;
