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;
reserve p,q for oriented Chain of G;

theorem Th26:
  v <> (the Source of G).(p.1) & v in vertices(p) implies ex i st
  1<=i & i <= len p & v = (the Target of G).(p.i)
proof
  set FT=the Target of G, FS=the Source of G;
  assume that
A1: v <> FS.(p.1) and
A2: v in vertices p;
  consider u being Vertex of G such that
A3: v=u and
A4: ex i st i in dom p & u in vertices(p/.i) by A2;
  consider i such that
A5: i in dom p and
A6: u in vertices(p/.i) by A4;
A7: u=FS.(p/.i) or u=FT.(p/.i) by A6,TARSKI:def 2;
A8: 1<=i by A5,FINSEQ_3:25;
A9: i <= len p by A5,FINSEQ_3:25;
  per cases by A3,A7,A8,A9,FINSEQ_4:15;
  suppose
A10: v=FT.(p.i);
    take i;
    thus thesis by A5,A10,FINSEQ_3:25;
  end;
  suppose
A11: v=FS.(p.i);
    consider j being Nat such that
A12: i=1+j by A8,NAT_1:10;
    reconsider j as Element of NAT by ORDINAL1:def 12;
A13: j < len p by A9,A12,NAT_1:13;
    take j;
    i > 1 by A1,A8,A11,XXREAL_0:1;
    then j >= 1 by A12,NAT_1:13;
    hence thesis by A11,A12,A13,GRAPH_1:def 15;
  end;
end;
