 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 Th29:
  for x,x1,x2,y1,y2 be Surreal holds
   (1_No +(x2 - x) * y2)*x1 +- (1_No +(x1 - x) * y1) * x2 ==
       (x1-x2)*(1_No-x*y1) + (y1-y2)*x1*(x-x2) &
   (1_No +(x2  - x) * y2)*x1 - (1_No +(x1  - x) * y1) * x2 ==
       (x1-x2)*(1_No-x*y2) + (y2-y1)*x2*(x1-x)
proof
  let x,x1,x2,y1,y2 be Surreal;
  (x2 + - x) * y2*x1 == (x2*y2 + (- x) * y2)*x1 == x2*y2*x1 + (- x) * y2*x1
  by SURREALR:51,SURREALR:67;
  then (x2 + - x) * y2*x1 == x2*y2*x1 + (- x) * y2*x1 by SURREALO:4;
  then (1_No +(x2 + - x) * y2)*x1 == 1_No*x1+(x2 + - x) * y2*x1
  == x1 + (x2*y2*x1 + (- x) * y2*x1) by SURREALR:43,67;
  then
A1:(1_No +(x2 + - x) * y2)*x1 == x1 + (x2*y2*x1 + (- x) * y2*x1)
  by SURREALO:4;
  (x1 + - x) * y1*x2 == (x1*y1 + (- x) * y1)*x2 == x1*y1*x2 + (- x) * y1*x2
  by SURREALR:51,SURREALR:67;
  then (x1 + - x) * y1*x2 == x1*y1*x2 + (- x) * y1*x2 by SURREALO:4;
  then (1_No +(x1 + - x) * y1)*x2 == 1_No*x2+(x1 + - x) * y1*x2
  == x2 + (x1*y1*x2 + (- x) * y1*x2) by SURREALR:43,67;
  then (1_No +(x1 + - x) * y1)*x2 == x2 + (x1*y1*x2 + (- x) * y1*x2)
  by SURREALO:4;
  then
A2: -((1_No +(x1 + - x) * y1)*x2) == -(x2 + (x1*y1*x2 + (- x) * y1*x2))
  by SURREALR:10;
A3:(x1+-x2)*1_No = x1+-x2;
  (x1+-x2)*(-x*y1) == x1*(-x*y1)+(-x2)*(-x*y1) by SURREALR:67;
  then (x1+-x2)*(1_No+-x*y1)==(x1+-x2)+ (x1+-x2)*(-x*y1) ==
  (x1+-x2)+( x1*(-x*y1)+(-x2)*(-x*y1))
  by A3,SURREALR:43,SURREALR:67;
  then
A4: (x1+-x2)*(1_No+-x*y1) == (x1+-x2)+( x1*(-x*y1)+(-x2)*(-x*y1))
  by SURREALO:4;
  set X3 = x1*(-x*y1),X4=(-x2)*(-x*y1),Y3=y1*x1*x,X5=(-y2)*x1 *x,
  X6=y1*x1*(-x2),X7=(-y2)*x1*(-x2);
A5: (y1*x1+(-y2)*x1)*x == (y1*x1*x)+(-y2)*x1 *x &
  (y1*x1+(-y2)*x1)*(-x2) == y1*x1*(-x2)+(-y2)*x1*(-x2) by SURREALR:67;
  (y1+-y2)*x1*(x+-x2) == (y1*x1+(-y2)*x1)*(x+-x2)==
  (y1*x1+(-y2)*x1)*x + (y1*x1+(-y2)*x1)*(-x2) by SURREALR:51,SURREALR:67;
  then (y1+-y2)*x1*(x+-x2) ==
  (y1*x1+(-y2)*x1)*x + (y1*x1+(-y2)*x1)*(-x2)==
  ((y1*x1*x)+(-y2)*x1 *x)+(y1*x1*(-x2)+(-y2)*x1*(-x2))
  by A5,SURREALO:4,SURREALR:43;
  then (y1+-y2)*x1*(x+-x2) ==
  ((y1*x1*x)+(-y2)*x1 *x)+(y1*x1*(-x2)+(-y2)*x1*(-x2)) by SURREALO:4;
  then
A6: (x1+-x2)*(1_No+-x*y1) + (y1+-y2)*x1*(x+-x2) ==
  (x1+-x2)+(X3+X4)+((Y3+X5)+(X6+X7)) by A4,SURREALR:43;
A7: (X3+X4)+((Y3+X5)+(X6+X7)) = X4+(X3+((Y3+X5)+(X6+X7))) by SURREALR:37
  .= (X3+(Y3+X5)+(X7+X6))+X4 by SURREALR:37
  .= (((X3+Y3)+X5)+(X7+X6))+X4 by SURREALR:37
  .= (((X3+Y3)+X5)+X7+X6)+X4 by SURREALR:37
  .= ((X3+Y3)+(X5 +X7)+X6)+X4 by SURREALR:37
  .= (X3+Y3)+(X5 +X7)+(X6+X4) by SURREALR:37;
A8: (x1+-x2)+(X3+X4)+((Y3+X5)+(X6+X7))
  =(x1+-x2)+( (X3+X4)+((Y3+X5)+(X6+X7))) by SURREALR:37
  .=(x1+-x2)+( (X3+Y3)+(X5 +X7))+(X6+X4) by A7,SURREALR:37
  .=x1+(( (X3+Y3)+(X5 +X7)) + -x2)+(X6+X4) by SURREALR:37
  .=x1+((( (X3+Y3)+(X5 +X7)) + -x2)+(X6+X4)) by SURREALR:37
  .=x1+(( (X3+Y3)+(X5 +X7)) + (-x2+(X6+X4))) by SURREALR:37
  .=x1+( (X3+Y3)+(X5 +X7)) + (-x2+(X6+X4)) by SURREALR:37;
A9: -(x1*(x*y1))== - Y3 by SURREALR:65,69;
  X3==-Y3 by A9,SURREALR:58;
  then X3+Y3 == (-Y3) + Y3 = Y3 - Y3 == 0_No by SURREALR:39,43;
  then
A10:X3+Y3 == 0_No by SURREALO:4;
  X5= (-(y2*x1)) *x by SURREALR:58
  .= - ((y2*x1) *x) by SURREALR:58
  .= (y2*x1) *(-x) by SURREALR:58;
  then
A11: X5 ==(-x)*y2*x1 by SURREALR:69;
  X7=(-(y2*x1))*(-x2) by SURREALR:58
  .=x2*(y2*x1) by SURREALR:58;
  then X7==x2*y2*x1 by SURREALR:69;
  then X5 +X7 == x2*y2*x1 + (- x) * y2*x1 by A11,SURREALR:43;
  then
A12: (X3+Y3)+(X5 +X7) == 0_No +(x2*y2*x1 + (- x) * y2*x1)
  = x2*y2*x1 + (- x) * y2*x1 by A10,SURREALR:43;
  X4 = ((-x)*y1)*(-x2) by SURREALR:58
  .=-(-x)*y1*x2 by SURREALR:58;
  then
A13: -x2+(X6+X4) ==-x2 +(-(x1*y1*x2) + - (- x) * y1*x2) by SURREALR:58;
  x1 + (x2*y2*x1 + (- x) * y2*x1) == x1+ ((X3+Y3)+(X5 +X7)) by A12,SURREALR:43;
  then
A14: x1 + (x2*y2*x1 + (- x) * y2*x1) +
  (-x2 +(-(x1*y1*x2) + - (- x) * y1*x2))
  == x1+( (X3+Y3)+(X5 +X7)) + (-x2+(X6+X4)) by A13,SURREALR:43;
  -( x2 + (x1*y1*x2 + (- x) * y1*x2)) = -x2 +-(x1*y1*x2 + (- x) * y1*x2)
  by SURREALR:40
  .= -x2 +(-(x1*y1*x2) + - (- x) * y1*x2) by SURREALR:40;
  then
A15: x1 + (x2*y2*x1 + (- x) * y2*x1) +
  (-x2 +(-(x1*y1*x2) + - (- x) * y1*x2))==
  (1_No +(x2 + - x) * y2)*x1+ -((1_No +(x1 + - x) * y1)*x2)
  by A1,A2,SURREALR:43;
  then (1_No +(x2 + - x) * y2)*x1+ -((1_No +(x1 + - x) * y1)*x2) ==
  x1+( (X3+Y3)+(X5 +X7)) + (-x2+(X6+X4)) by A14,SURREALO:4;
  hence (1_No +(x2 - x) * y2)*x1 +- (1_No +(x1 - x) * y1) * x2 ==
  (x1-x2)*(1_No-x*y1) + (y1-y2)*x1*(x-x2) by A6,A8,SURREALO:4;
A16:(x1+-x2)*1_No = x1+-x2;
  (x1+-x2)*(-x*y2) == x1*(-x*y2)+(-x2)*(-x*y2) by SURREALR:67;
  then (x1+-x2)*(1_No+-x*y2)==(x1+-x2)+ (x1+-x2)*(-x*y2) ==
  (x1+-x2)+( x1*(-x*y2)+(-x2)*(-x*y2)) by A16,SURREALR:43,SURREALR:67;
  then
A17:(x1+-x2)*(1_No+-x*y2)==(x1+-x2)+( x1*(-x*y2)+(-x2)*(-x*y2))
  by SURREALO:4;
A18: (y2*x2+(-y1)*x2)*x1 == y2*x2*x1+(-y1)*x2*x1 &
  (y2*x2+(-y1)*x2)*(-x) == y2*x2*(-x)+(-y1)*x2*(-x) by SURREALR:67;
  (y2+-y1)*x2*(x1+-x) == (y2*x2+(-y1)*x2)*(x1+-x)==
  (y2*x2+(-y1)*x2)*x1 + (y2*x2+(-y1)*x2)*(-x) by SURREALR:51,67;
  then (y2+-y1)*x2*(x1+-x) ==
  (y2*x2+(-y1)*x2)*x1 + (y2*x2+(-y1)*x2)*(-x)
  == (y2*x2*x1+(-y1)*x2*x1)+( y2*x2*(-x)+(-y1)*x2*(-x))
  by A18,SURREALO:4,SURREALR:43;
  then
A19: (y2+-y1)*x2*(x1+-x) ==
  (y2*x2*x1+(-y1)*x2*x1)+( y2*x2*(-x)+(-y1)*x2*(-x)) by SURREALO:4;
  set A2=y2*x2*x1, A3 = x1*(-x*y2), A4 = -x2, A5 = (-y1)*x2*x1,
  A6 = (-y1)*x2*(-x),A7 = (-x2)*(-x*y2), B7=y2*x2*(-x);
A20: (x1+-x2)*(1_No+-x*y2) + (y2+-y1)*x2*(x1+-x) ==
  (x1+A4)+(A3+A7)+((A2+A5)+(B7+A6)) by A19,A17,SURREALR:43;
  (A7+A3)+(B7+A2) = (A3+A7)+B7+A2 by SURREALR:37
  .= A3+(A7+B7)+A2 by SURREALR:37
  .= (A2+A3)+(A7+B7) by SURREALR:37;
  then ((A3+A7)+(A2+B7+(A5+A6))) = (A2+A3)+(A7+B7)+(A5+A6) by SURREALR:37;
  then
A21:(x1+A4)+((A3+A7)+(A2+B7+(A5+A6)))
  =x1+((A2+A3)+(A7+B7) +(A5+A6)+A4) by SURREALR:37
  .=x1+((A2+A3)+(A7+B7) +((A5+A6)+A4)) by SURREALR:37
  .=x1+((A2+A3)+(A7+B7)) +(A4+(A5+A6)) by SURREALR:37
  .=x1+(A2+A3)+(A7+B7) +(A4+(A5+A6)) by SURREALR:37;
A22:(A2+A5)+(B7+A6) = (A2+A5)+A6+B7 by SURREALR:37
  .= A2+(A5+A6)+B7 by SURREALR:37
  .= A2+B7+(A5+A6) by SURREALR:37;
A23: (x1+A4)+(A3+A7)+(A2+B7+(A5+A6)) =
  x1+(A2+A3)+(A7+B7) +(A4+(A5+A6)) by A21,SURREALR:37;
A24: A2+A3 = x2*y2*x1 + (- x) * y2*x1 by SURREALR:58;
  A5 = (-(y1*x2))*x1 by SURREALR:58
  .= -(y1*x2*x1) by SURREALR:58;
  then
A25:A5 == -(x1*y1*x2) by SURREALR:65,69;
A26: (-y1)*x2*(-x) = (-(y1*x2))*(-x) by SURREALR:58
  .= -((y1*x2)*(-x)) by SURREALR:58;
  (-y1)*x2*(-x) == - ((- x) * y1*x2) by A26,SURREALR:65,69;
  then A5+A6== -(x1*y1*x2) + - (- x) * y1*x2 by A25,SURREALR:43;
  then
A27: A4+(A5+A6)== (-x2 +(-(x1*y1*x2) + - (- x) * y1*x2)) by SURREALR:43;
A28: A7  = - (x2*(-x*y2)) by SURREALR:58;
  -x*y2 = (-x)*y2 by SURREALR:58;
  then x2*(-x*y2)  == y2*x2*(-x) by SURREALR:69;
  then A7 == - B7 by A28,SURREALR:10;
  then A7+B7 == -B7+B7= B7 - B7 == 0_No by SURREALR:39,43;
  then
A29: A7+B7  == 0_No by SURREALO:4;
  x1+(A2+A3)+(A7+B7)==x1 + (x2*y2*x1 + (- x) * y2*x1)+0_No
  by A29,SURREALR:43,A24;
  then x1+(A2+A3)+(A7+B7) +(A4+(A5+A6))==
  x1 + (x2*y2*x1 + (- x) * y2*x1) +(-x2 +(-(x1*y1*x2) + - (- x) * y1*x2))
  by A27,SURREALR:43;
  then (x1+A4)+(A3+A7)+(A2+B7+(A5+A6))==
  (1_No +(x2 + - x) * y2)*x1+ -((1_No +(x1 + - x) * y1)*x2)
  by A23,SURREALO:4,A15;
  hence thesis by A22,A20,SURREALO:4;
end;
