 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 Th30:
  for x1,y1,Ix1 be Surreal st x1*Ix1==1_No holds
     x1*y +x*y1 - x1*y1 == 1_No +x1*( y - (1_No +(x1-x)*y1)*Ix1)
proof
  let x1,y1,Ix1 be Surreal such that
A1: x1*Ix1==1_No;
  (x1 * ( (1_No +(x1+-x)*y1)*Ix1)) == (1_No +(x1+-x)*y1)*(x1*Ix1) ==
  (1_No +(x1+-x)*y1)*1_No = 1_No +(x1+-x)*y1
  by A1,SURREALR:69,51;
  then (x1 * ( (1_No +(x1+-x)*y1)*Ix1)) == 1_No +(x1+-x)*y1
  by SURREALO:4;
  then
A2: - (x1 * ( (1_No +(x1+-x)*y1)*Ix1)) == -(1_No +(x1+-x)*y1)
  = -1_No +- ((x1+-x)*y1) by SURREALR:10,40;
  -((x1+-x)*y1) == -(x1*y1 + (-x)*y1) by SURREALR:65,SURREALR:67;
  then -1_No +- ((x1+-x)*y1) == -1_No + -(x1*y1 + (-x)*y1) by SURREALR:43;
  then
A3: - (x1 * ( (1_No +(x1+-x)*y1)*Ix1)) ==
  -1_No + -(x1*y1 + (-x)*y1) by A2,SURREALO:4;
A4: (x1) * (- (1_No +(x1+-x)*y1)*Ix1) == -1_No + -(x1*y1 + (-x)*y1)
  by A3,SURREALR:58;
  x1*( y +- (1_No +(x1+-x)*y1)*Ix1) == (x1* y) + x1 *(- (1_No +(x1+-x)*y1)*Ix1)
  == (x1* y) + (-1_No + -(x1*y1 + (-x)*y1)) by A4,SURREALR:43,67;
  then x1*( y +- (1_No +(x1+-x)*y1)*Ix1) ==
  (x1* y) + (-1_No + -(x1*y1 + (-x)*y1)) by SURREALO:4;
  then
A5:1_No +x1*( y +- (1_No +(x1+-x)*y1)*Ix1) ==
  1_No + ((x1* y) + (-1_No + -(x1*y1 + (-x)*y1))) by SURREALR:43;
A6: 1_No -1_No==0_No by SURREALR:39;
  - ((-x)*y1) = - - (x*y1) by SURREALR:58
  .= x*y1;
  then
A7: (x1* y) + -(x1*y1 + (-x)*y1) = (x1* y) +( -(x1*y1) + x*y1 )
  by SURREALR:40
  .= x1*y +x*y1 +- x1*y1 by SURREALR:37;
  1_No + ((x1* y) + (-1_No + -(x1*y1 + (-x)*y1))) =
  1_No + (-1_No + ((x1* y) + -(x1*y1 + (-x)*y1))) by SURREALR:37
  .= (1_No + -1_No) + ((x1* y) + -(x1*y1 + (-x)*y1)) by SURREALR:37;
  then 1_No + ((x1* y) + (-1_No + -(x1*y1 + (-x)*y1))) ==
  0_No + (x1*y +x*y1 +- x1*y1) = x1*y +x*y1 +- x1*y1 by A7,A6,SURREALR:43;
  hence thesis by A5,SURREALO:4;
end;
