reserve A,B for Ordinal,
        o for object,
        x,y,z for Surreal,
        n for Nat,
        r,r1,r2 for Real;

theorem Th83:
  for r be Sequence of REAL
    for y,s be Sequence
       for A,B be Ordinal st A c= B holds
         x in_meets_terms s,y,r,A iff x in_meets_terms s,y|B,r|B,A
proof
  let r be Sequence of REAL;
  let y,s be Sequence;
  let alpha,B be Ordinal such that
A1: alpha c= B;
  thus x in_meets_terms s,y,r,alpha implies x in_meets_terms s,y|B,r|B,alpha
  proof
    assume
A2: x in_meets_terms s,y,r,alpha;
    let beta be Ordinal,sb,yb be Surreal such that
A3: beta in alpha & sb = s.beta & yb = (y|B).beta;
    (r|B).beta = r.beta & yb = y.beta by A1,A3,FUNCT_1:49;
    hence thesis by A2,A3;
  end;
  assume
A4:x in_meets_terms s,y|B,r|B,alpha;
  let beta be Ordinal,sb,yb be Surreal such that
A5:beta in alpha & sb = s.beta & yb = y.beta;
  (r|B).beta = r.beta & yb = (y|B).beta by A5,A1,FUNCT_1:49;
  hence thesis by A4,A5;
end;
