reserve S, R for 1-sorted,
  X for Subset of R,
  T for TopStruct,
  x for set;
reserve H for non empty multMagma,
  P, Q, P1, Q1 for Subset of H,
  h for Element of H;
reserve G for Group,
  A, B for Subset of G,
  a for Element of G;

theorem Th14:
  (the multF of H).:[:P,Q:] = P*Q
proof
  set f = the multF of H;
  hereby
    let y be object;
    assume y in f.:[:P,Q:];
    then consider x being object such that
    x in [:the carrier of H,the carrier of H:] and
A1: x in [:P,Q:] and
A2: y = f.x by FUNCT_2:64;
    consider a, b being object such that
A3: a in P & b in Q and
A4: x = [a,b] by A1,ZFMISC_1:def 2;
    reconsider a, b as Element of H by A3;
    y = a*b by A2,A4;
    hence y in P*Q by A3;
  end;
  let y be object;
  assume y in P*Q;
  then consider g, h being Element of H such that
A5: y = g*h and
A6: g in P & h in Q;
  [g,h] in [:P,Q:] by A6,ZFMISC_1:87;
  hence thesis by A5,FUNCT_2:35;
end;
