reserve E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  i, j, m, n for Nat,
  p for Point of TOP-REAL 2;

theorem
  for D be non empty set for f be FinSequence of D st 1 <= i & i < len f
  holds mid(f,i,len f-'1)^<*f/.(len f)*> = mid(f,i,len f)
proof
  let D be non empty set;
  let f be FinSequence of D;
  assume that
A1: 1 <= i and
A2: i < len f;
A3: i+1 <= len f by A2,NAT_1:13;
  then
A4: i+1-1 <= len f-1 by XREAL_1:9;
  then
A5: i <= len f-'1 by XREAL_0:def 2;
  0+i <= len f-1 by A4;
  then len f-1-i >= 0 by XREAL_1:19;
  then
A6: len f-'1-i >= 0 by A4,XREAL_0:def 2;
A7: len f-i >= 0 by A3,XREAL_1:19;
  len f <= len f+1 by NAT_1:11;
  then len f-1 <= len f+1-1 by XREAL_1:9;
  then
A8: len f-'1 <= len f by XREAL_0:def 2;
  then
A9: len mid(f,i,len f-'1) + 1 = len f-'1-'i+1+1 by A1,A5,FINSEQ_6:186
    .= len f-'1-i+1+1 by A6,XREAL_0:def 2
    .= len f-1-i+1+1 by A4,XREAL_0:def 2
    .= len f-'i+1 by A7,XREAL_0:def 2
    .= len mid(f,i,len f) by A1,A2,FINSEQ_6:186;
A10: now
    1 < len f by A1,A2,XXREAL_0:2;
    then len f in Seg len f by FINSEQ_1:1;
    then
A11: len f in dom f by FINSEQ_1:def 3;
    i in Seg len f by A1,A2,FINSEQ_1:1;
    then
A12: i in dom f by FINSEQ_1:def 3;
    let j be Nat;
    assume that
A13: 1 <= j and
A14: j <= len mid(f,i,len f);
    per cases by A14,XXREAL_0:1;
    suppose
      j < len mid(f,i,len f);
      then
A15:  j <= len mid(f,i,len f-'1) by A9,NAT_1:13;
      then j in Seg len mid(f,i,len f-'1) by A13,FINSEQ_1:1;
      then
A16:  j in dom mid(f,i,len f-'1) by FINSEQ_1:def 3;
      1 <= len f-'1 by A1,A5,XXREAL_0:2;
      then len f-'1 in Seg len f by A8,FINSEQ_1:1;
      then
A17:  len f-'1 in dom f by FINSEQ_1:def 3;
      j in Seg len mid(f,i,len f) by A13,A14,FINSEQ_1:1;
      then
A18:  j in dom mid(f,i,len f) by FINSEQ_1:def 3;
      j in NAT by ORDINAL1:def 12;
      hence
      (mid(f,i,len f-'1)^<*f/.(len f)*>)/.j = mid(f,i,len f-'1)/.j by A13,A15,
BOOLMARK:7
        .= f/.(j+i-'1) by A5,A12,A16,A17,SPRECT_2:3
        .= mid(f,i,len f)/.j by A2,A12,A11,A18,SPRECT_2:3;
    end;
    suppose
A19:  j = len mid(f,i,len f);
      hence (mid(f,i,len f-'1)^<*f/.(len f)*>)/.j = f/.(len f) by A9,
FINSEQ_4:67
        .= mid(f,i,len f)/.j by A12,A11,A19,SPRECT_2:9;
    end;
  end;
  len (mid(f,i,len f-'1)^<*f/.(len f)*>) = len mid(f,i,len f-'1) + 1 by
FINSEQ_2:16;
  hence thesis by A9,A10,FINSEQ_5:13;
end;
