 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 Th37:
  for X1,X2,Y1,Y2 be surreal-membered set st
    X2 <=_ X1 & Y2 <=_ Y1 & [X1,Y1] is surreal
  holds [X2,Y2] is surreal
proof
  let X1,X2,Y1,Y2 be surreal-membered set such that
A1: X2 <=_ X1 & Y2 <=_ Y1 & [X1,Y1] is surreal;
A2: X2 << Y2
  proof
    let l,r be Surreal such that
A3: l in X2 & r in Y2;
    consider l1,l2 be Surreal such that
A4: l1 in X1 & l2 in X1 & l1 <= l <= l2 by A3,A1;
    consider r1,r2 be Surreal such that
A5: r1 in Y1 & r2 in Y1 & r1 <= r <= r2 by A3,A1;
    X1 =L_[X1,Y1] <<R_[X1,Y1]=Y1 by A1,SURREAL0:45;
    then l < r1 by A4,A5,SURREALO:4;
    hence thesis by A5,SURREALO:4;
  end;
  consider M be Ordinal such that
A6:for o st o in X2 \/ Y2 ex A be Ordinal st A in M & o in Day A
  by SURREAL0:47;
  [X2,Y2] in Day M by A6,A2,SURREAL0:46;
  hence thesis;
end;
