reserve x,X,Y for set;
reserve g,r,r1,r2,p,p1,p2 for Real;
reserve R for Subset of REAL;
reserve seq,seq1,seq2,seq3 for Real_Sequence;
reserve Ns for increasing sequence of NAT;
reserve n for Nat;
reserve W for non empty set;
reserve h,h1,h2 for PartFunc of W,REAL;

theorem Th1:
  seq1=seq2-seq3 iff for n holds seq1.n=seq2.n-seq3.n
proof
  thus seq1=seq2-seq3 implies for n holds seq1.n=seq2.n-seq3.n
  proof
    assume
A1: seq1=seq2-seq3;
    now
      let n;
      seq1.n=seq2.n+(-seq3).n by A1,SEQ_1:7;
      then seq1.n=seq2.n+-seq3.n by SEQ_1:10;
      hence seq1.n=seq2.n-seq3.n;
    end;
    hence thesis;
  end;
  assume
A2: for n holds seq1.n=seq2.n-seq3.n;
  now
    let n;
    thus seq1.n = seq2.n-seq3.n by A2
      .= seq2.n+-seq3.n
      .= seq2.n+(-seq3).n by SEQ_1:10;
  end;
  hence thesis by SEQ_1:7;
end;
