reserve A,B for Ordinal,
        o for object,
        x,y,z for Surreal,
        n for Nat,
        r,r1,r2 for Real;

theorem Th70:
  x == y & not x==0_No implies
     omega-y x = omega-y y & omega-r x = omega-r y
proof
  assume
A1: x == y & not x==0_No;
A2: not y==0_No by A1,SURREALO:4;
  |.x.| is positive by A1,Th36;
  then
A3: |.x.|,|.y.| are_commensurate by A1,Th48,Th8;
  then
A4: omega-y x = omega-y y by A2,A1,Th61;
  set rx= omega-r x;
  |.x - No_omega^ omega-y x * uReal.rx .| infinitely< |.x.| by A1,Def8;
  then
A5: |.x +- No_omega^ omega-y y * uReal.rx .| infinitely< |.y.|
  by A4,A3, Th16;
  x +- No_omega^ omega-y y * uReal.rx ==
  y +- No_omega^ omega-y y * uReal.rx by SURREALR:43,A1;
  then |.y - No_omega^ omega-y y * uReal.rx .| infinitely< |.y.|
  by A5,Th17,Th48;
  hence thesis by A2,Def8,A3,A1,Th61;
end;
