 reserve A,B,O for Ordinal,
      n,m for Nat,
      a,b,o for object,
      x,y,z for Surreal,
      X,Y,Z for set,
      Inv,I1,I2 for Function;

theorem Th28:
   for x1,x1R,y1,y1R be Surreal st 0_No < x1 & x1 * x1R == 1_No &
     0_No < y1 & y1 * y1R == 1_No & x * y1 < y * x1
   holds x * x1R < y * y1R
proof
  let x1,x1R,y1,y1R be Surreal such that
A1:0_No < x1 & x1 * x1R == 1_No &
  0_No < y1 & y1 * y1R == 1_No & x * y1 < y * x1;
A2: 0_No <= x1 & 0_No <= y1 by A1;
  1_No is positive;
  then 0_No < x1 * x1R & 0_No < y1 * y1R by A1,SURREALO:4;
  then
A3: 0_No < x1R & 0_No < y1R by A2,SURREALR:72;
  then x * y1 * y1R < y * x1 * y1R by A1,SURREALR:70;
  then
A4: x * y1 * y1R * x1R  < y * x1 * y1R *x1R by A3,SURREALR:70;
  x * y1 * y1R == x * (y1 * y1R)== x *1_No =x by SURREALR:51,SURREALR:69,A1;
  then x * y1 * y1R==x by SURREALO:4;
  then
A5: x * y1 * y1R * x1R == x * x1R by SURREALR:51;
  y * x1 * y1R *x1R == y * (x1 * y1R) *x1R == y * ((x1 * y1R) *x1R)
  by SURREALR:51,SURREALR:69;
  then y * x1 * y1R *x1R == y * ((x1 * y1R) *x1R) == y * (y1R*(x1 *x1R))
  by SURREALR:69,SURREALO:4,SURREALR:51;
  then
A6:y * x1 * y1R *x1R== y * (y1R*(x1 *x1R)) by SURREALO:4;
  y1R*(x1 *x1R) == y1R *1_No =y1R by SURREALR:51,A1;
  then y * (y1R*(x1 *x1R))==y*y1R by SURREALR:51;
  then y * x1 * y1R *x1R == y*y1R by A6,SURREALO:4;
  then x * y1 * y1R * x1R < y*y1R by A4,SURREALO:4;
  hence thesis by A5,SURREALO:4;
end;
