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

theorem Th79:
  for r be Sequence of REAL
    for y,s be Sequence
      for alpha be Ordinal holds
  x in_meets_terms s,y,r,alpha iff x in_meets_terms (s|succ alpha),y,r,alpha
proof
  let r be Sequence of REAL;
  let y,s be Sequence;
  let alpha be Ordinal;
  thus x in_meets_terms s,y,r,alpha implies
  x in_meets_terms s|succ alpha,y,r,alpha
  proof
    assume
A1: x in_meets_terms s,y,r,alpha;
    let beta be Ordinal,sb,yb be Surreal such that
A2: beta in alpha & sb = (s|succ alpha).beta & yb = y.beta;
    beta in succ alpha by A2,ORDINAL1:8;
    then (s|succ alpha).beta = s.beta by FUNCT_1:49;
    hence thesis by A1,A2;
  end;
  assume
A3:x in_meets_terms s|succ alpha,y,r,alpha;
  let beta be Ordinal,sb,yb be Surreal such that
A4:beta in alpha & sb = s.beta & yb = y.beta;
  beta in succ alpha by A4,ORDINAL1:8;
  then (s|succ alpha).beta = s.beta by FUNCT_1:49;
  hence thesis by A3,A4;
end;
