reserve p, q for FinSequence,
  X, Y, x, y, e for set,
  D for non empty set,
  i, j, k, l, m, n, r for Nat;
reserve G for Graph;
reserve v, v1, v2, v3, v4 for Element of G;
reserve vs, vs1, vs2 for FinSequence of the carrier of G,
  c, c1, c2 for Chain of G;

theorem Th42:
  1<=m & m<=n & n<=len c & c1 = (m,n)-cut c & vs is_vertex_seq_of
  c & vs1 = (m,n+1)-cut vs implies vs1 is_vertex_seq_of c1
proof
  assume that
A1: 1<=m and
A2: m<=n and
A3: n<=len c;
A4: m<=n+1 by A2,NAT_1:12;
  assume
A5: c1 = (m,n)-cut c;
  then
A6: len c1 +m= n+1 by A1,A3,A4,Lm2;
  assume that
A7: vs is_vertex_seq_of c and
A8: vs1 = (m,n+1)-cut vs;
A9: len vs = len c + 1 by A7;
  then
A10: n+1<=len vs by A3,XREAL_1:6;
  then len vs1 +m = n+1+1 by A1,A8,A4,FINSEQ_6:def 4;
  hence
A11: len vs1 = len c1 + 1 by A6;
  let k be Nat;
  assume that
A12: 1<=k and
A13: k<=len c1;
  0+1<=k by A12;
  then consider j such that
  0<=j and
A14: j<len c1 and
A15: k=j+1 by A13,FINSEQ_6:127;
  set i = m+j;
  j<len vs1 by A11,A14,NAT_1:13;
  then
A16: vs1.k = vs.i by A1,A8,A4,A10,A15,FINSEQ_6:def 4;
  m+k<=len c1 +m by A13,XREAL_1:7;
  then m+j+1-1<=len c1 +m-1 by A15,XREAL_1:9;
  then
A17: i<=len c by A3,A6,XXREAL_0:2;
  then i<=len vs by A9,NAT_1:12;
  then
A18: vs/.i=vs.i by A1,FINSEQ_4:15,NAT_1:12;
A19: k<=len c1 by A14,A15,NAT_1:13;
  then
A20: k<=len vs1 by A11,NAT_1:12;
  1<=k+1 by NAT_1:12;
  then
A21: vs1/.(k+1)=vs1.(k+1) by A11,A19,FINSEQ_4:15,XREAL_1:7;
  0+1=1;
  then 1<=k by A15,NAT_1:13;
  then
A22: vs1/.k=vs1.k by A20,FINSEQ_4:15;
  set v2 = vs/.(i+1);
  set v1 = vs/.i;
A23: m+j+1 = m+(j+1);
  1<=i by A1,NAT_1:12;
  then
A24: c.i joins v1, v2 by A7,A17;
  1<=i+1 by NAT_1:12;
  then
A25: vs/.(i+1)=vs.(i+1) by A9,A17,FINSEQ_4:15,XREAL_1:7;
  j+1<len vs1 by A11,A14,XREAL_1:6;
  then vs1.(k+1) = vs.(i+1) by A1,A8,A4,A10,A15,A23,FINSEQ_6:def 4;
  hence thesis by A1,A3,A5,A4,A14,A15,A24,A16,A18,A25,A22,A21,Lm2;
end;
