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

theorem CutAssoc:
  for A, C being non empty set,
      L being associative BinOp of A,
      LC being BinOp of C st
    C c= A & LC = L || C holds
     LC is associative
  proof
    let A, C be non empty set;
    let L be associative BinOp of A,
        LC be BinOp of C;
    assume
Z1: C c= A & LC = L || C;
A1: dom LC = [:C,C:] by FUNCT_2:def 1;
    for a, b, c being Element of C holds
      LC.(a,LC.(b,c)) = LC.(LC.(a,b),c)
    proof
      let a, b, c be Element of C;
      reconsider aa = a, bb = b, cc = c as Element of A by Z1;
W2:   LC.(aa,bb) = L.(a,b) by ZFMISC_1:87,FUNCT_1:49,Z1;
ZZ:   L.(aa,L.(bb,cc)) = L.(L.(aa,bb),cc) by BINOP_1:def 3;
      set y = [aa,LC.[bb,cc]];
      thus LC.(a,LC.(b,c)) = L.y by FUNCT_1:49,Z1,ZFMISC_1:87
           .= L.(L.(aa,bb),cc) by ZZ,ZFMISC_1:87,FUNCT_1:49,Z1
           .= LC.(LC.(a,b),c) by FUNCT_1:47,A1,Z1,W2,ZFMISC_1:87;
    end;
    hence thesis by BINOP_1:def 3;
  end;
