reserve U1,U2,U3 for Universal_Algebra,
  n,m for Nat,
  x,y,z for object,
  A,B for non empty set,
  h1 for FinSequence of [:A,B:];
reserve h1 for homogeneous quasi_total non empty PartFunc of
    (the carrier of U1)*,the carrier of U1,
  h2 for homogeneous quasi_total non empty PartFunc of
    (the carrier of U2)*,the carrier of U2;

theorem Th5:
  U1,U2 are_similar implies
    for o1 be operation of U1,o2 be operation of U2,
        o be operation of [:U1,U2:], n be Nat st
    o1 = (the charact of U1).n & o2 = (the charact of U2).n &
    o = (the charact of [:U1,U2:]).n & n in dom the charact of(U1) holds
      arity o = arity o1 & arity o = arity o2 & o = [[:o1,o2:]]
proof
  assume
A1: U1,U2 are_similar;
A2: dom Opers(U1,U2) = Seg len Opers(U1,U2) by FINSEQ_1:def 3;
A3: dom the charact of(U1) = Seg len the charact of(U1) by FINSEQ_1:def 3;
  let o1 be operation of U1,o2 be operation of U2,o be operation of [:U1,U2:],
  n be Nat;
  assume that
A4: o1 = (the charact of U1).n and
A5: o2 = (the charact of U2).n and
A6: o = (the charact of [:U1,U2:]).n and
A7: n in dom the charact of(U1);
A8: dom (signature U1) = Seg len (signature U1) & len signature U1 = len
  the charact of (U1) by FINSEQ_1:def 3,UNIALG_1:def 4;
  then
A9: (signature U1).n = arity o1 by A4,A7,A3,UNIALG_1:def 4;
A10: signature U1 = signature U2 by A1; then
A11: (signature U2).n = arity o2 by A5,A7,A3,A8,UNIALG_1:def 4;
A12: [:U1,U2:] = UAStr (# [:the carrier of U1,the carrier of U2:], Opers(U1,
    U2) #) & len the charact of(U1) = len Opers(U1,U2) by A1,Def4,Def5;
  then o = [[:o1,o2:]] by A1,A4,A5,A6,A7,A3,A2,Def4;
  then
A13: dom o = (arity o1)-tuples_on [:the carrier of U1,the carrier of U2:] by
A10,A9,A11,Def3;
  (ex x being FinSequence st x in dom o) & for x be FinSequence st x in
  dom o holds len x = arity o1
  proof
    set a = the Element of (arity o1)-tuples_on
      [:the carrier of U1,the carrier of U2:];
    a in dom o by A13;
    hence ex x being FinSequence st x in dom o;
    let x be FinSequence;
    assume x in dom o;
    then ex s be Element of ([:the carrier of U1,the carrier of U2:]) * st s =
    x & len s = arity o1 by A13;
    hence thesis;
  end;
  hence thesis by A1,A4,A5,A6,A7,A3,A2,A12,A9,A11,Def4,MARGREL1:def 25;
end;
