
theorem Th30:
  for P being _finite non _trivial Path-like _Graph holds P.minDegree() = 1
proof
  let P be _finite non _trivial Path-like _Graph;
  consider p being non empty _finite Path-like Graph-yielding FinSequence
  such that
    A1: p.1 is 2-vertex Path-like & p.len p = P & len p + 1 = P.order() and
    A2: for n being Element of dom p st n <= len p - 1
      ex v1,v2 being Vertex of P, e being object
      st p.(n+1) is addAdjVertex of p.n,v1,e,v2 &
        e in the_Edges_of P \ the_Edges_of p.n &
        ((v1 in Endvertices(p.n) & not v2 in the_Vertices_of p.n) or
          (not v1 in the_Vertices_of p.n & v2 in Endvertices(p.n))) by Th25;
  defpred P[Nat] means for H being _Graph st H = p.($1+1) & $1 <= len p - 1
    holds H.minDegree() = 1;
  A3: P[0]
  proof
    let H be _Graph;
    assume H = p.(0+1) & 0 <= len p - 1;
    then A4: H is 2-vertex Path-like by A1;
    now
      take v = the Vertex of H;
      thus A6: v.degree() = 1 by A4, GLIB_000:174;
      let w be Vertex of H;
      thus v.degree() c= w.degree() by A4, A6, GLIB_000:174;
    end;
    hence thesis by GLIB_013:36;
  end;
  A7: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume A8: P[k];
    let H be _Graph;
    assume A9: H = p.(k+1+1) & k+1 <= len p - 1;
    len p - 1 <= len p - 0 by XREAL_1:10;
    then A10: k+1 <= len p by A9, XXREAL_0:2;
    0+1 <= k+1 by XREAL_1:6;
    then reconsider n = k+1 as Element of dom p by A10, FINSEQ_3:25;
    consider v1, v2 being Vertex of P, e being object such that
      A11: H is addAdjVertex of p.n,v1,e,v2 and
      A12: e in the_Edges_of P \ the_Edges_of p.n and
      A13: ((v1 in Endvertices(p.n) & not v2 in the_Vertices_of p.n) or
        (not v1 in the_Vertices_of p.n & v2 in Endvertices(p.n))) by A2, A9;
    k+1-1 <= len p - 1 by A10, XREAL_1:9;
    then A14: (p.n).minDegree() = 1 by A8;
    A15: not e in the_Edges_of p.n by A12, XBOOLE_0:def 5;
    per cases by A13; :: symmetric cases
    suppose A16: v1 in Endvertices(p.n) & not v2 in the_Vertices_of p.n;
      then reconsider v1 as Vertex of p.n;
      now
        A17: the_Vertices_of H = (the_Vertices_of p.n) \/ {v2}
          by A11, A15, A16, GLIB_006:def 12;
        v2 in {v2} by TARSKI:def 1;
        then reconsider v2 as Vertex of H by A17, XBOOLE_0:def 3;
        take v2;
        thus A18: v2.degree() = 1 by A11, A15, A16, GLIB_006:141, GLIB_000:174;
        let w be Vertex of H;
        per cases;
        suppose w = v2;
          hence v2.degree() c= w.degree();
        end;
        suppose w <> v2;
          then not w in {v2} by TARSKI:def 1;
          then reconsider u = w as Vertex of p.n by A17, XBOOLE_0:def 3;
          A19: 1 c= u.degree() by A14, GLIB_013:35;
          p.n is Subgraph of H by A11, GLIB_006:57;
          then u.degree() c= w.degree() by Th1;
          hence v2.degree() c= w.degree() by A18, A19, XBOOLE_1:1;
        end;
      end;
      hence thesis by GLIB_013:36;
    end;
    suppose A20: not v1 in the_Vertices_of p.n & v2 in Endvertices(p.n);
      then reconsider v2 as Vertex of p.n;
      now
        A21: the_Vertices_of H = (the_Vertices_of p.n) \/ {v1}
          by A11, A15, A20, GLIB_006:def 12;
        v1 in {v1} by TARSKI:def 1;
        then reconsider v1 as Vertex of H by A21, XBOOLE_0:def 3;
        take v1;
        thus A22: v1.degree() = 1 by A11, A15, A20, GLIB_006:142, GLIB_000:174;
        let w be Vertex of H;
        per cases;
        suppose w = v1;
          hence v1.degree() c= w.degree();
        end;
        suppose w <> v1;
          then not w in {v1} by TARSKI:def 1;
          then reconsider u = w as Vertex of p.n by A21, XBOOLE_0:def 3;
          A23: 1 c= u.degree() by A14, GLIB_013:35;
          p.n is Subgraph of H by A11, GLIB_006:57;
          then u.degree() c= w.degree() by Th1;
          hence v1.degree() c= w.degree() by A22, A23, XBOOLE_1:1;
        end;
      end;
      hence thesis by GLIB_013:36;
    end;
  end;
  A24: for k being Nat holds P[k] from NAT_1:sch 2(A3,A7);
  0 < len p;
  then 0+1 < len p + 1 by XREAL_1:6;
  then 1 <= len p by NAT_1:13;
  then 1-1 <= len p - 1 by XREAL_1:9;
  then len p -' 1 + 1 = len p by NAT_D:72;
  hence thesis by A1, A24;
end;
