theorem Th72:
  F is associative & F is having_an_inverseOp & F is having_a_unity &
  F.:(f,f9) = C-->the_unity_wrt F implies
  f = (the_inverseOp_wrt F)*f9 & (the_inverseOp_wrt F)*f = f9
proof
  assume that
A1: F is associative & F is having_an_inverseOp & F is having_a_unity and
A2: F.:(f,f9) = C-->the_unity_wrt F;
  set u = the_inverseOp_wrt F;
  set e = the_unity_wrt F;
  reconsider g = C-->e as Function of C,D;
  now
    let c;
    F.(f.c,f9.c) = g.c by A2,FUNCOP_1:37
      .= e;
    hence f.c = u.(f9.c) by A1,Th60
      .= (u*f9).c by FUNCT_2:15;
  end;
  hence f = (the_inverseOp_wrt F)*f9 by FUNCT_2:63;
  now
    let c;
    F.(f.c,f9.c) = g.c by A2,FUNCOP_1:37
      .= e;
    then f9.c = u.(f.c) by A1,Th60;
    hence (u*f).c = f9.c by FUNCT_2:15;
  end;
  hence thesis by FUNCT_2:63;
end;
