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
  U1,U2 are_similar implies [:U1,U2:],U1 are_similar
proof
A1: dom signature(U1) = Seg len signature(U1) by FINSEQ_1:def 3;
A2: dom signature(U1) = Seg len signature(U1) by FINSEQ_1:def 3;
A3: dom the charact of([:U1,U2:]) = Seg len the charact of([:U1,U2:]) by
FINSEQ_1:def 3;
A4: dom the charact of(U2) = Seg len the charact of(U2) by FINSEQ_1:def 3;
A5: dom the charact of(U1) = Seg len the charact of(U1) by FINSEQ_1:def 3;
  assume
A6: U1,U2 are_similar;
  then
  [:U1,U2:] = UAStr (# [:the carrier of U1,the carrier of U2:], Opers(U1,
    U2) #) by Def5;
  then len the charact of([:U1,U2:]) = len the charact of(U1) by A6,Def4;
  then
A7: len signature ([:U1,U2:]) = len the charact of(U1) by UNIALG_1:def 4
    .= len signature(U1) by UNIALG_1:def 4;
A8: dom signature([:U1,U2:]) = Seg len signature([:U1,U2:]) by FINSEQ_1:def 3;
  now
    let n be Nat;
    assume
A9: n in dom signature(U1);
    then n in dom the charact of([:U1,U2:]) by A7,A1,A3,UNIALG_1:def 4;
    then reconsider
    o12 = (the charact of [:U1,U2:]).n as operation of [:U1,U2:] by
FUNCT_1:def 3;
    len signature(U1) = len signature(U2) by A6
      .= len the charact of(U2) by UNIALG_1:def 4;
    then reconsider
    o2 = (the charact of U2).n as operation of U2 by A1,A4,A9,FUNCT_1:def 3;
A10: o2 = (the charact of U2).n;
A11: n in Seg len the charact of(U1) by A2,A9,UNIALG_1:def 4;
    then n in dom the charact of (U1) by FINSEQ_1:def 3;
    then reconsider o1 = (the charact of U1).n as operation of U1 by
FUNCT_1:def 3;
    (signature(U1)).n = arity(o1) & (signature([:U1,U2:])).n = arity(o12)
    by A7,A1,A8,A9,UNIALG_1:def 4;
    hence (signature(U1)).n = (signature([:U1,U2:])).n by A6,A5,A11,A10,Th5;
  end;
  hence thesis by A7,FINSEQ_2:9;
end;
