reserve G for Go-board,
  i,j,k,m,n for Nat;
reserve f,g,g1,g2 for FinSequence of TOP-REAL 2;

theorem Th34:
  f is unfolded & g is unfolded & f/.len f = g/.1 & LSeg(f,len f
  -' 1) /\ LSeg(g,1) = { f/.len f } implies f^'g is unfolded
proof
  assume that
A1: f is unfolded and
A2: g is unfolded and
A3: f/.len f = g/.1 and
A4: LSeg(f,len f -' 1) /\ LSeg(g,1) = { f/.len f };
  set c = (1+1, len g)-cut g, k = len f -' 1;
  reconsider g9 = g as unfolded FinSequence of TOP-REAL 2 by A2;
A5: c = g9/^1 by Th13;
A6: f ^' g = f^(1+1, len g)-cut g by FINSEQ_6:def 5;
  len g <= 2 implies len g = 0 or ... or len g = 2;
  then per cases;
  suppose
    f is empty;
    hence thesis by A6,A5,FINSEQ_1:34;
  end;
  suppose
    len g = 0;
    then g = {};
    then c = <*>the carrier of TOP-REAL 2 by A5,FINSEQ_6:80;
    hence thesis by A1,A6,FINSEQ_1:34;
  end;
  suppose
    len g = 1;
    then c = {} by A5,FINSEQ_6:167;
    hence thesis by A1,A6,FINSEQ_1:34;
  end;
  suppose that
A7: f is non empty and
A8: len g = 1+1;
A9: k+1 = len f by A7,NAT_1:14,XREAL_1:235;
    g|len g = g by FINSEQ_1:58;
    then g = <*g/.1,g/.2*> by A8,FINSEQ_5:81;
    then
A10: f^'g = f^<*g/.2*> by A6,A5,FINSEQ_6:46;
    1 <= len g - 1 by A8;
    then 1 <= len(g/^1) by A8,RFINSEQ:def 1;
    then
A11: 1 in dom(g/^1) by FINSEQ_3:25;
    then
A12: c/.1 = (g/^1).1 by A5,PARTFUN1:def 6
      .= g.(1+1) by A8,A11,RFINSEQ:def 1
      .= g/.(1+1) by A8,FINSEQ_4:15;
    then LSeg(g,1) = LSeg(f/.len f,c/.1) by A3,A8,TOPREAL1:def 3;
    hence thesis by A1,A4,A9,A10,A12,SPPOL_2:30;
  end;
  suppose that
A13: f is non empty and
A14: 2 < len g;
A15: 1 <= len g by A14,XXREAL_0:2;
    then
A16: LSeg(g/^1,1) = LSeg(g,1+1) by SPPOL_2:4;
    1+1 <= len g by A14;
    then 1 <= len g - 1 by XREAL_1:19;
    then 1 <= len(g/^1) by A15,RFINSEQ:def 1;
    then
A17: 1 in dom(g/^1) by FINSEQ_3:25;
    then
A18: c/.1 = (g/^1).1 by A5,PARTFUN1:def 6
      .= g.(1+1) by A15,A17,RFINSEQ:def 1
      .= g/.(1+1) by A14,FINSEQ_4:15;
    then
A19: LSeg(g,1) = LSeg(f/.len f,c/.1) by A3,A14,TOPREAL1:def 3;
A20: k+1 = len f by A13,NAT_1:14,XREAL_1:235;
    1 + 2 <= len g by A14,NAT_1:13;
    then LSeg(g,1) /\ LSeg(c,1) = {c/.1} by A5,A18,A16,TOPREAL1:def 6;
    hence thesis by A1,A4,A6,A5,A20,A19,SPPOL_2:31;
  end;
end;
