 reserve n,m for Nat,
      o for object,
      p for pair object,
      x,y,z for Surreal;

theorem
  0_No <= x <= y implies sqrt x <= sqrt y
proof
  assume that
A1:0_No <= x <= y and
A2:sqrt y < sqrt x;
A3:0_No <= sqrt x & sqrt x * sqrt x == x by A1,Th19;
  0_No <= y by A1,SURREALO:4;
  then
A4: 0_No <= sqrt y & sqrt y * sqrt y == y by Th19;
  0_No < sqrt x by A4,A2,SURREALO:4;
  then sqrt y* sqrt x < sqrt x * sqrt x by A2,SURREALR:70;
  then
A5: sqrt y* sqrt x < x by A3,SURREALO:4;
  sqrt y <= sqrt x by A2;
  then sqrt y* sqrt y <= sqrt x * sqrt y by A4,SURREALR:75;
  then y <= sqrt x * sqrt y by A4,SURREALO:4;
  hence thesis by A1,A5,SURREALO:4;
end;
