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;
reserve v,v1,v2,v3 for Element of G;

theorem Th23:
  len pe = 1 implies vertices(pe) = vertices(pe/.1)
proof
  set FS=the Source of G, FT=the Target of G;
  assume
A1: len pe = 1;
  then
A2: 1 in dom pe by FINSEQ_3:25;
  now
    let x be object;
    hereby
      assume x in vertices(pe);
      then consider y being Vertex of G such that
A3:   y=x and
A4:   ex i st i in dom pe & y in vertices(pe/.i);
      consider i such that
A5:   i in dom pe and
A6:   y in vertices(pe/.i) by A4;
      1<=i & i <= len pe by A5,FINSEQ_3:25;
      hence x in vertices(pe/.1) by A1,A3,A6,XXREAL_0:1;
    end;
    assume
A7: x in vertices(pe/.1);
    then x=FS.(pe/.1) or x=FT.(pe/.1) by TARSKI:def 2;
    then x=FS.(pe.1) or x=FT.(pe.1) by A1,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,A7;
    hence x in vertices(pe);
  end;
  hence thesis by TARSKI:2;
end;
