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

theorem Th49:
  |. |.x.| - |.y.| .| <= |.x - y.|
proof
  y-y == 0_No by SURREALR:39;
  then x = x+0_No == x+(y+-y) = x+ -y +y
  by SURREALR:43,SURREALR:37;
  then |.x.| == |.x+ -y + y.| <= |.x+ -y.| +|.y.| by Th37,Th48;
  then |.x.| <= |.x+ -y.| +|.y.| by SURREALO:4;
  then
A1: |.x.| - |.y.| <= |.x+ -y.| by SURREALR:42;
  x-x == 0_No by SURREALR:39;
  then y = y+0_No == y+(x+-x) = y+ -x +x by SURREALR:43,37;
  then |.y.| == |.y+ -x + x.| <= |.y+ -x.| +|.x.| by Th37,Th48;
  then
A2: |.y.| <= |.y+ -x.| +|.x.| by SURREALO:4;
  - (x +-y) = -x + - - y by SURREALR:40
  .= -x + y;
  then
  |.y+ -x.| = |.-(x +-y).| == |. x +-y.| by Th39,Th38;
  then |.y+ -x.| +|.x.| == |.x +- y.| + |.x.| by SURREALR:43;
  then |.y.| <= |.x +- y.| + |.x.| by A2,SURREALO:4;
  then - |.x - y.| - |.x.| = -(|.x - y.| + |.x.|) <= - |.y.|
  by SURREALR:40,SURREALR:10;
  then - |.x - y.| <= |.x.| - |.y.| by SURREALR:42;
  hence thesis by A1,Th35;
end;
