
theorem Th19:
for S being Ring,
    R being Subring of S
for x being Element of S, x1 being Element of R st x = x1 holds -x = -x1
proof
let R be Ring, S being Subring of R;
let x be Element of R, x1 be Element of S;
set C = the carrier of R, C1 = the carrier of S, a = -x1;
assume A1: x = x1;
C1 c= C by C0SP1:def 3;
then reconsider g = a as Element of R;
x + g = x1 + a by A1,Th17 .= 0.S by VECTSP_1:19 .= 0.R by C0SP1:def 3;
hence thesis by VECTSP_1:16;
end;
