 reserve n,i for Nat;
 reserve p for Prime;

theorem CutComm:
  for A, C being non empty set,
      L being commutative BinOp of A,
      LC being BinOp of C st
    C c= A & LC = L || C holds
     LC is commutative
  proof
    let A, C be non empty set;
    let L be commutative BinOp of A,
        LC be BinOp of C;
    assume
Z1: C c= A & LC = L || C;
    for a, b being Element of C holds LC.(a,b) = LC.(b,a)
    proof
      let a, b be Element of C;
      reconsider aa = a, bb = b as Element of A by Z1;
ZZ:   L.(aa,bb) = L.(bb,aa) by BINOP_1:def 2;
      thus LC.(a,b) = L.[aa,bb] by ZFMISC_1:87,FUNCT_1:49,Z1
         .= LC.(b,a) by ZZ,ZFMISC_1:87,FUNCT_1:49,Z1;
    end;
    hence thesis by BINOP_1:def 2;
  end;
