theorem
  F is associative implies F[;](F.(x1,x2),f) = F[;](x1,F[;](x2,f))
proof
  assume
A1: F is associative;
  per cases;
  suppose
    Y = {};
    hence thesis;
  end;
  suppose
A2: Y <> {};
    now
      let y;
      reconsider x3 = f.y as Element of X by A2,FUNCT_2:5;
      thus (F[;](F.(x1,x2),f)).y = F.(F.(x1,x2),f.y) by A2,Th53
        .= F.(x1,F.(x2,x3)) by A1
        .= F.(x1,(F[;](x2,f)).y) by A2,Th53;
    end;
    hence thesis by A2,Th54;
  end;
end;
