
theorem
  for g1 being FinSequence of TOP-REAL 2, i being Nat
  st 1 <= i & i <= len g1 & g1 is being_S-Seq holds
  g1/.1 in L~mid(g1, i, len g1) implies i = 1
proof
  let g1 be FinSequence of TOP-REAL 2, i be Nat;
  assume that
A1: 1 <= i and
A2: i <= len g1 and
A3: g1 is being_S-Seq;
  assume g1/.1 in L~mid(g1, i, len g1);
  then consider j being Nat such that
A4: 1 <= j and
A5: j+1 <= len mid(g1, i, len g1) and
A6: g1/.1 in LSeg(mid(g1, i, len g1),j) by SPPOL_2:13;
A7: j+1 in dom mid(g1, i, len g1) by A4,A5,SEQ_4:134;
A8: mid(g1, i, len g1) = g1/^(i-'1) by A2,FINSEQ_6:117;
  j <= j+1 by NAT_1:11;
  then
A9: j <= len (g1/^(i-'1)) by A5,A8,XXREAL_0:2;
  then
A10: j in dom (g1/^(i-'1)) by A4,FINSEQ_3:25;
  i-'1 <= i by A1,Th1;
  then
A11: i-'1 <= len g1 by A2,XXREAL_0:2;
  then
A12: j <= len g1 - (i-'1) by A9,RFINSEQ:def 1;
  j <= i-'1+j by NAT_1:11;
  then
A13: 1 <= i-'1+j by A4,XXREAL_0:2;
A14: j+(i-'1) <= len g1 by A12,XREAL_1:19;
  then
A15: i-'1+j in dom g1 by A13,FINSEQ_3:25;
A16: (g1/^(i-'1))/.j = (g1/^(i-'1)).j by A10,PARTFUN1:def 6
    .= g1.(i-'1+j) by A4,A14,FINSEQ_6:114
    .= g1/.(i-'1+j) by A15,PARTFUN1:def 6;
A17: 1 <= j+1 by NAT_1:11;
  j+1 <= i-'1+(j+1) by NAT_1:11;
  then
A18: 1 <= i-'1+(j+1) by A17,XXREAL_0:2;
  j+1 <= len (g1/^(i-'1)) by A2,A5,FINSEQ_6:117;
  then
A19: j+1 <= len g1 - (i-'1) by A11,RFINSEQ:def 1;
A20: 1 <= j+1 by A7,FINSEQ_3:25;
A21: j+1+(i-'1) <= len g1 by A19,XREAL_1:19;
  then
A22: i-'1+(j+1) in dom g1 by A18,FINSEQ_3:25;
  j+1 in dom (g1/^(i-'1)) by A2,A7,FINSEQ_6:117;
  then
A23: (g1/^(i-'1))/.(j+1) = (g1/^(i-'1)).(j+1) by PARTFUN1:def 6
    .= g1.(i-'1+(j+1)) by A20,A21,FINSEQ_6:114
    .= g1/.(i-'1+(j+1)) by A22,PARTFUN1:def 6;
A24: i-'1+j+1 = i-'1+(j+1);
A25: i-'1+j+1 <= len g1 by A21;
A26: LSeg(mid(g1, i, len g1),j) =
  LSeg (g1/.(i-'1+j), g1/.(i-'1+(j+1)) ) by A4,A5,A8,A16,A23,TOPREAL1:def 3
    .= LSeg ( g1, i-'1+j ) by A13,A21,A24,TOPREAL1:def 3;
A27: 1+1 <= len g1 by A3,TOPREAL1:def 8;
  then g1/.1 in LSeg ( g1, 1 ) by TOPREAL1:21;
  then
A28: g1/.1 in LSeg ( g1, 1 ) /\ LSeg ( g1, i-'1+j ) by A6,A26,XBOOLE_0:def 4;
  then
A29: LSeg (g1, 1) meets LSeg ( g1, i-'1+j );
  assume i <> 1;
  then 1 < i by A1,XXREAL_0:1;
  then 1+1 <= i by NAT_1:13;
  then 2-'1 <= i-'1 by NAT_D:42;
  then 2-'1+1 <= i-'1+j by A4,XREAL_1:7;
  then
A30: i-'1+j >= 2 by XREAL_1:235;
A31: g1 is s.n.c. unfolded one-to-one by A3;
A32: 1 in dom g1 by A27,SEQ_4:134;
A33: 1+1 in dom g1 by A27,SEQ_4:134;
  per cases by A30,XXREAL_0:1;
  suppose i-'1+j = 2;
    then g1/.1 in { g1/.(1+1) } by A25,A28,A31,TOPREAL1:def 6;
    then g1/.1 = g1/.(1+1) by TARSKI:def 1;
    hence thesis by A31,A32,A33,PARTFUN2:10;
  end;
  suppose i-'1+j > 2;
    then i-'1+j > 1+1;
    hence thesis by A29,A31,TOPREAL1:def 7;
  end;
end;
