theorem Th71:
  F is having_a_unity & F is associative & F is having_an_inverseOp
  implies F.:(f,(the_inverseOp_wrt F)*f) = C-->the_unity_wrt F &
  F.:((the_inverseOp_wrt F)*f,f) = C-->the_unity_wrt F
proof
  set u = the_inverseOp_wrt F;
  reconsider g = C-->the_unity_wrt F as Function of C,D;
  assume
A1: F is having_a_unity & F is associative & F is having_an_inverseOp;
  now
    let c;
    thus (F.:(f,u*f)).c = F.(f.c,(u*f).c) by FUNCOP_1:37
      .= F.(f.c,u.(f.c)) by FUNCT_2:15
      .= the_unity_wrt F by A1,Th59
      .= g.c;
  end;
  hence F.:(f,(the_inverseOp_wrt F)*f) = C-->the_unity_wrt F by FUNCT_2:63;
  now
    let c;
    thus (F.:(u*f,f)).c = F.((u*f).c,f.c) by FUNCOP_1:37
      .= F.(u.(f.c),f.c) by FUNCT_2:15
      .= the_unity_wrt F by A1,Th59
      .= g.c;
  end;
  hence thesis by FUNCT_2:63;
end;
