
theorem lemalph:
-alpha = alpha & alpha" = alpha-1 & alpha" <> alpha
proof
J: Char(Z/2) = 2 by RING_3:def 6;
Z/2 is Subring of embField(canHomP X^2+X+1) by FIELD_4:def 1; then
I: Char embField(canHomP X^2+X+1) = 2 by J,RING_3:89;
H: 2 '*' alpha = 2 '*' (1.embField(canHomP X^2+X+1) * alpha)
              .= (2 '*' 1.embField(canHomP X^2+X+1)) * alpha by REALALG2:5
              .= 0.embField(canHomP X^2+X+1) * alpha by I,REALALG2:24; then
alpha + alpha = 0.embField(canHomP X^2+X+1) by RING_5:2; then
0.embField(canHomP X^2+X+1) - alpha
   = alpha + (alpha - alpha) by RLVECT_1:def 3
  .= alpha + 0.embField(canHomP X^2+X+1) by RLVECT_1:15;
hence H3: -alpha = alpha;
thus alpha" = alpha-1 by lemZ2,H3,RLVECT_1:31;
now assume alpha" = alpha; then
   alpha = -alpha + -1.embField(canHomP X^2+X+1) by RLVECT_1:31,lemZ2; then
   alpha + alpha
         = (alpha + -alpha) + -1.embField(canHomP X^2+X+1) by RLVECT_1:def 3
        .= 0.embField(canHomP X^2+X+1) + -1.embField(canHomP X^2+X+1)
           by RLVECT_1:5; then
   -0.embField(canHomP X^2+X+1) = --1.embField(canHomP X^2+X+1) by H,RING_5:2;
   hence contradiction;
   end;
hence thesis;
end;
