reserve i,j,k,l for natural Number;
reserve A for set, a,b,x,x1,x2,x3 for object;
reserve D,D9,E for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve d9,d19,d29,d39 for Element of D9;
reserve p,q,r for FinSequence;
reserve s for Element of D*;

theorem
  for p1,p2,q being FinSequence st p1 c= q & p2 c= q & len p1 = len p2
  holds p1 = p2
proof
  let p1,p2,q be FinSequence;
  assume that
A1: p1 c= q and
A2: p2 c= q and
A3: len p1 = len p2;
  reconsider i = len p1 as Element of NAT;
A4: dom p1 = Seg i & dom p2 = Seg i by A3,FINSEQ_1:def 3;
  now
    let j be Nat;
    assume
A5: j in dom p1;
    hence p1.j = q.j by A1,GRFUNC_1:2
      .= p2.j by A2,A4,A5,GRFUNC_1:2;
  end;
  hence thesis by A4;
end;
