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

theorem Th66:
  x is positive & r <> 0 implies
    |. uReal.r * x .|, x are_commensurate
proof
  assume
A1:x is positive & r <> 0;
  then per cases;
  suppose
A2: 0 < r;
    then 0_No <= uReal.r * x by A1,SURREALI:def 8;
    then uReal.r * x = |. uReal.r * x.| by Def6;
    hence thesis by Lm10,A1,A2;
  end;
  suppose
A3: r < 0;
    0_No <= (uReal.-r) * x by A3,A1,SURREALI:def 8;
    then
A4: (uReal.-r) * x = |.(uReal.-r) * x.| by Def6;
    (uReal.-r) *x == (-uReal.r)*x == - (uReal.r*x) by SURREALN:56,
    SURREALR:51,SURREALR:58;
    then (uReal.-r) *x == - (uReal.r*x) by SURREALR:58;
    then |.(uReal.-r) * x.| == |.- (uReal.r*x).| == |. uReal.r*x .|
    by Th48,Th39,Th38;
    then (uReal.-r) * x == |. uReal.r*x .| by A4,SURREALO:4;
    hence thesis by A3,Lm10,A1,Th5;
  end;
end;
