
theorem Th73:
  for A,B,C1,C2 being category, E1 being Functor of C1 [x] A,B,
      E2 being Functor of C2 [x] A,B st E1 is covariant & E2 is covariant &
  C1,E1 is_exponent_of A,B & C2,E2 is_exponent_of A,B holds C1 ~= C2
  proof
    let A,B,C1,C2 be category;
    let E1 be Functor of C1 [x] A,B;
    let E2 be Functor of C2 [x] A,B;
    assume
A1: E1 is covariant;
    assume
A2: E2 is covariant;
    assume
A3: C1,E1 is_exponent_of A,B;
    assume
A4: C2,E2 is_exponent_of A,B;
    ex F being Functor of C1,C2, G being Functor of C2,C1 st
    F is covariant & G is covariant & G (*) F = id C1 & F (*) G = id C2
    proof
      consider F be Functor of C1,C2 such that
A5:   F is covariant & E1 = E2 (*)(F [x] id A) & for H1 being Functor of C1,C2
      st H1 is covariant & E1 = E2 (*)(H1 [x] id A) holds F = H1
      by A1,A2,A4,Def34;
      consider G be Functor of C2,C1 such that
A6:   G is covariant & E2 = E1 (*)(G [x] id A) & for H1 being Functor of C2,C1
      st H1 is covariant & E2 = E1 (*)(H1 [x] id A) holds G = H1
      by A1,A2,A3,Def34;
      take F,G;
      thus F is covariant & G is covariant by A5,A6;
      consider H2 be Functor of C1,C1 such that
A7:  H2 is covariant & E1 = E1 (*)(H2 [x] id A) & for H1 being Functor of C1,C1
      st H1 is covariant & E1 = E1 (*)(H1 [x] id A) holds H2 = H1
      by A1,A3,Def34;
A8:   G [x] id A is covariant by A6,Def22;
A9:   F [x] id A is covariant by A5,Def22;
      E1 = E1 (*) ((G [x] id A)(*)(F [x] id A)) by A5,A6,A8,A9,A1,CAT_7:10
      .= E1 (*)(G(*)F [x]( id A)(*)(id A)) by A5,A6,Th50
      .= E1 (*)(G(*)F [x] id A) by CAT_7:11;
      then
A10:  G(*)F = H2 by A7,A5,A6,CAT_6:35;
      E1 = E1(*)id(C1 [x] A) by A1,CAT_7:11
      .= E1(*)(id C1 [x] id A) by Th51;
      hence G(*)F = id C1 by A7,A10;
      consider H3 be Functor of C2,C2 such that
A11:  H3 is covariant & E2 = E2(*)(H3 [x] id A) & for H1 being Functor of C2,C2
      st H1 is covariant & E2 = E2 (*) (H1 [x] id A) holds H3 = H1
      by A2,A4,Def34;
      E2 = E2 (*) ((F [x] id A)(*)(G [x] id A)) by A2,A5,A6,A8,A9,CAT_7:10
      .= E2 (*)(F(*)G [x] (id A)(*)(id A)) by A5,A6,Th50
      .= E2 (*)(F(*)G [x] id A) by CAT_7:11;
      then
A12:  F(*)G = H3 by A11,A5,A6,CAT_6:35;
      E2 = E2(*)id(C2 [x] A) by A2,CAT_7:11
      .= E2(*)(id C2 [x] id A) by Th51;
      hence F(*)G = id C2 by A11,A12;
    end;
    hence C1,C2 are_isomorphic by CAT_6:def 28;
  end;
