
theorem Th5:
  for D being non empty set, S being non empty Subset of D, f1,f2
  being BinOp of D, g1,g2 being BinOp of S st g1 = f1||S & g2 = f2||S holds f1
  absorbs f2 implies g1 absorbs g2
proof
  let D be non empty set, S be non empty Subset of D, f1,f2 be BinOp of D;
  let g1,g2 be BinOp of S;
  assume that
A1: g1 = f1||S and
A2: g2 = f2||S;
  assume
A3: for a,b being Element of D holds f1.(a,f2.(a,b)) = a;
  let a,b be Element of S;
A4: dom g2 = [:S,S:] by FUNCT_2:def 1;
  dom g1 = [:S,S:] by FUNCT_2:def 1;
  hence g1.(a,g2.(a,b)) = f1.[a,g2.(a,b)] by A1,FUNCT_1:47
    .= f1.[a,f2.[a,b]] by A2,A4,FUNCT_1:47
    .= f1.(a,f2.(a,b))
    .= a by A3;
end;
