
theorem Th51:
  for n being non zero Nat, C1, C2 being n-vertex Cycle-like _Graph
  holds C2 is C1-isomorphic
proof
  let n being non zero Nat, C1, C2 being n-vertex Cycle-like _Graph;
  per cases;
  suppose n = 1;
    then A1: C1 is _trivial & C2 is _trivial;
    C1.size() = n & C2.size() = n by GLIB_013:def 4;
    then ex F being PGraphMapping of C1, C2 st F is Disomorphism
      by A1, GLIB_010:177;
    then C2 is C1-Disomorphic by GLIB_010:def 24;
    hence thesis;
  end;
  suppose n <> 1;
    then C1.order() <> 1 & C2.order() <> 1 by GLIB_013:def 3;
    then A2: C1 is non _trivial & C2 is non _trivial by GLIB_000:26;
    set e1 = the Edge of C1, e2 = the Edge of C2;
    set v1 = (the_Source_of C1).e1, w1 = (the_Target_of C1).e1;
    set v2 = (the_Source_of C2).e2, w2 = (the_Target_of C2).e2;
    A3: e1 DJoins v1,w1,C1 & e2 DJoins v2,w2,C2 by GLIB_000:def 14;
    consider P1 being non _trivial _finite Path-like _Graph such that
      A4: not e1 in the_Edges_of P1 & C1 is addEdge of P1,v1,e1,w1 and
      A5: Endvertices P1 = {v1,w1} by A2, A3, Th45;
    reconsider v1,w1 as Vertex of P1 by A4, Lm8;
    consider P2 being non _trivial _finite Path-like _Graph such that
      A6: not e2 in the_Edges_of P2 & C2 is addEdge of P2,v2,e2,w2 and
      A7: Endvertices P2 = {v2,w2} by A2, A3, Th45;
    reconsider v2,w2 as Vertex of P2 by A6, Lm8;
    P1.order() = C1.order() by A4, Lm8
      .= n by GLIB_013:def 3;
    then A8: P1 is n-vertex by GLIB_013:def 3;
    P2.order() = C2.order() by A6, Lm8
      .= n by GLIB_013:def 3;
    then P2 is n-vertex by GLIB_013:def 3;
    then P2 is P1-isomorphic by A8, Th32;
    then consider F0 being PGraphMapping of P1, P2 such that
      A9: F0 is isomorphism by GLIB_010:def 23;
    A10: dom F0_V = the_Vertices_of P1 by A9, GLIB_010:def 11;
    v1 in Endvertices P1 by A5, TARSKI:def 2;
    then 1 = v1.degree() by GLIB_006:56, GLIB_000:174
      .= (F0_V/.v1).degree() by A9, GLIBPRE0:93;
    then F0_V/.v1 in {v2,w2} by A7, GLIB_006:56, GLIB_000:174;
    then F0_V.v1 in {v2,w2} by A10, PARTFUN1:def 6;
    then A11: F0_V.v1 = v2 or F0_V.v1 = w2 by TARSKI:def 2;
    w1 in Endvertices P1 by A5, TARSKI:def 2;
    then 1 = w1.degree() by GLIB_006:56, GLIB_000:174
      .= (F0_V/.w1).degree() by A9, GLIBPRE0:93;
    then F0_V/.w1 in {v2,w2} by A7, GLIB_006:56, GLIB_000:174;
    then F0_V.w1 in {v2,w2} by A10, PARTFUN1:def 6;
    then A12: F0_V.w1 = v2 or F0_V.w1 = w2 by TARSKI:def 2;
    v1 <> w1
    proof
      assume v1 = w1;
      then Endvertices P1 = {v1} by A5, ENUMSET1:29;
      then card Endvertices P1 = 1 by CARD_1:30;
      hence contradiction by Th37;
    end;
    then A13: (F0_V.v1 = v2 & F0_V.w1 = w2) or
      (F0_V.v1 = w2 & F0_V.w1 = v2) by A9, A10, A11, A12, FUNCT_1:def 4;
    consider F being PGraphMapping of C1, C2 such that
      F = [F0_V, F0_E +* (e1 .--> e2)] and
      F0 is weak_SG-embedding implies F is weak_SG-embedding and
      A14: F0 is isomorphism implies F is isomorphism
      by A4, A6, A10, A13, GLIB_010:153;
    thus thesis by A9, A14, GLIB_010:def 23;
  end;
end;
