 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 Th35:
  x==y implies divset(X,x,Y) <=_ divset(X,y,Y)
proof
  assume
A1:x==y;
  let z be Surreal;
  assume z in divset(X,x,Y);
  then consider x1,y1 be Surreal such that
A2:  0_No < x1 & x1 in X & y1 in Y and
A3:  z = (1_No +(x1-x)*y1)* (x1") by Def15;
A4:(1_No +(x1-y)*y1)* (x1") in divset(X,y,Y) by A2,Def15;
  -x == -y by A1,SURREALR:10;
  then x1+-x == x1+-y by SURREALR:43;
  then (x1+-x)*y1 == (x1+-y)*y1 by SURREALR:51;
  then 1_No+(x1+-x)*y1 == 1_No+(x1+-y)*y1 by SURREALR:43;
  then z == (1_No+(x1+-y)*y1)*(x1") by A3,SURREALR:51;
  hence thesis by A4;
end;
