reserve n,m,i,j,k for Nat,
  x,y,e,X,V,U for set,
  W,f,g for Function;
reserve p,q for FinSequence;
reserve G for Graph,
  pe,qe for FinSequence of the carrier' of G;

theorem Th17:
  vertices pe c= V iff for i be Nat st i in dom pe holds vertices( pe/.i) c= V
proof
  set FS=the Source of G, FT=the Target of G;
  hereby
    assume
A1: vertices pe c= V;
    hereby
      let i be Nat;
      assume
A2:   i in dom pe;
      then
A3:   1<=i & i <= len pe by FINSEQ_3:25;
      thus vertices(pe/.i) c= V
      proof
        let x be object;
        assume
A4:     x in vertices(pe/.i);
        then x = FS.(pe/.i) or x=FT.(pe/.i) by TARSKI:def 2;
        then x = FS.(pe.i) or x=FT.(pe.i) by A3,FINSEQ_4:15;
        then reconsider y=x as Vertex of G by A2,FINSEQ_2:11,FUNCT_2:5;
        y in {v where v is Vertex of G : ex i st i in dom pe & v in
        vertices(pe/.i)} by A2,A4;
        hence thesis by A1;
      end;
    end;
  end;
  assume
A5: for i be Nat st i in dom pe holds vertices(pe/.i) c= V;
  let x be object;
  assume x in vertices pe;
  then consider y being Vertex of G such that
A6: y=x and
A7: ex i st i in dom pe & y in vertices(pe/.i);
  consider i such that
A8: i in dom pe and
A9: y in vertices(pe/.i) by A7;
  vertices(pe/.i) c= V by A5,A8;
  hence thesis by A6,A9;
end;
