reserve U0,U1,U2,U3 for Universal_Algebra,
  n for Nat,
  x,y for set;
reserve A for non empty Subset of U0,
  o for operation of U0,
  x1,y1 for FinSequence of A;

theorem Th6:
  for U0,U1 be Universal_Algebra, o0 be operation of U0, o1 be
operation of U1, n be Nat st U0 is SubAlgebra of U1 & n in dom the charact of(
U0) & o0 = (the charact of(U0)).n & o1 = (the charact of(U1)).n holds arity o0
  = arity o1
proof
  let U0,U1 be Universal_Algebra, o0 be operation of U0, o1 be operation of U1
  ,n;
  assume that
A1: U0 is SubAlgebra of U1 and
A2: n in dom the charact of(U0) & o0 = (the charact of(U0)).n and
A3: o1 = (the charact of(U1)).n;
  reconsider A =the carrier of U0 as non empty Subset of U1 by A1,Def7;
  A is opers_closed by A1,Def7;
  then
A4: A is_closed_on o1;
  n in dom Opers(U1,A) & o0 = Opers(U1,A).n by A1,A2,Def7;
  then o0 = o1/.A by A3,Def6;
  then o0 = o1|((arity o1)-tuples_on A) by A4,Def5;
  then dom o0 = dom o1 /\ ((arity o1)-tuples_on A) by RELAT_1:61;
  then
A5: dom o0 = ((arity o1)-tuples_on the carrier of U1) /\ ((arity o1)
  -tuples_on A) by MARGREL1:22
    .= (arity o1)-tuples_on A by MARGREL1:21;
  dom o0 =(arity o0)-tuples_on A by MARGREL1:22;
  hence thesis by A5,FINSEQ_2:110;
end;
