reserve G,G1,G2 for _Graph;
reserve W,W1,W2 for Walk of G;
reserve e,x,y,z for set;
reserve v for Vertex of G;
reserve n,m for Element of NAT;

theorem Th132:
  W2 is non trivial & W2.edges() c= W1.edges() implies W2
  .vertices() c= W1.vertices()
proof
  assume that
A1: W2 is non trivial and
A2: W2.edges() c= W1.edges();
A3: 3 <= len W2 by A1,Lm54;
  now
    let v be object;
    assume v in W2.vertices();
    then consider n being odd Element of NAT such that
A4: n <= len W2 and
A5: W2.n = v by Lm45;
    now
      per cases;
      suppose
        n = len W2;
        then 3-1 < n-0 by A3,XREAL_1:15;
        then reconsider n5 = n-2*1 as odd Element of NAT by INT_1:5;
A6:     1 <= n5+1 by NAT_1:12;
        n5 < n - 0 by XREAL_1:15;
        then
A7:     n5 < len W2 by A4,XXREAL_0:2;
        then
A8:     W2.(n5+1) Joins W2.n5, W2.(n5+2), G by Def3;
        n5+1 <= len W2 by A7,NAT_1:13;
        then W2.(n5+1) in W2.edges() by A6,Lm46;
        then consider m being even Element of NAT such that
A9:     1 <= m and
A10:    m <= len W1 and
A11:    W1.m = W2.(n5+1) by A2,Lm46;
        reconsider maa1 = m - 1 as odd Element of NAT by A9,INT_1:5;
A12:    maa1 < len W1 - 0 by A10,XREAL_1:15;
        then
A13:    W1.(maa1+1) Joins W1.maa1, W1.(maa1+2), G by Def3;
A14:    W1.maa1 = W1.vertexAt(maa1) by A12,Def8;
A15:    maa1+2 <= len W1 by A12,Th1;
        then W1.(maa1+2) = W1.vertexAt(maa1+2) by Def8;
        then v = W1.vertexAt(maa1) or v = W1.vertexAt(maa1+2) by A5,A8,A11,A13
,A14,GLIB_000:15;
        hence v in W1.vertices() by A12,A15,Th87;
      end;
      suppose
        n <> len W2;
        then
A16:    n < len W2 by A4,XXREAL_0:1;
        then W2.(n+1) in W2.edges() by Th98;
        then consider m being even Element of NAT such that
A17:    1 <= m and
A18:    m <= len W1 and
A19:    W1.m = W2.(n+1) by A2,Lm46;
A20:    W1.m Joins v, W2.(n+2),G by A5,A16,A19,Def3;
        reconsider maa1 = m - 1 as odd Element of NAT by A17,INT_1:5;
A21:    maa1 < len W1 - 0 by A18,XREAL_1:15;
        then
A22:    W1.(maa1+1) Joins W1.maa1, W1.(maa1+2), G by Def3;
A23:    W1.maa1 = W1.vertexAt(maa1) by A21,Def8;
A24:    maa1+2 <= len W1 by A21,Th1;
        then W1.(maa1+2) = W1.vertexAt(maa1+2) by Def8;
        then v = W1.vertexAt(maa1) or v = W1.vertexAt(maa1+2)
          by A20,A22,A23,GLIB_000:15;
        hence v in W1.vertices() by A21,A24,Th87;
      end;
    end;
    hence v in W1.vertices();
  end;
  hence thesis by TARSKI:def 3;
end;
