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 Th9:
  for p,q be FinSequence of TOP-REAL 2 for v be Point of TOP-REAL
  2 st p is_in_the_area_of q holds Rotate(p,v) is_in_the_area_of q
proof
  let p,q be FinSequence of TOP-REAL 2;
  let v be Point of TOP-REAL 2;
  assume
A1: p is_in_the_area_of q;
  per cases;
  suppose
A2: v in rng p;
    now
      let n;
      assume n in dom Rotate(p,v);
      then n in dom p by FINSEQ_6:180;
      then
A3:   n in Seg len p by FINSEQ_1:def 3;
      then
A4:   n <= len p by FINSEQ_1:1;
A5:   0+1 <= n by A3,FINSEQ_1:1;
      then
A6:   n-1 >= 0 by XREAL_1:19;
      per cases;
      suppose
A7:     n <= len(p:-v);
        then n <= len p-v..p+1 by A2,FINSEQ_5:50;
        then n-1 <= len p-v..p by XREAL_1:20;
        then n-1+v..p <= len p by XREAL_1:19;
        then
A8:     n-'1+v..p <= len p by A6,XREAL_0:def 2;
        1 <= v..p by A2,FINSEQ_4:21;
        then 0+1 <= n-'1+v..p by XREAL_1:7;
        then n-'1+v..p in Seg len p by A8,FINSEQ_1:1;
        then
A9:     n-'1+v..p in dom p by FINSEQ_1:def 3;
A10:    Rotate(p,v)/.n = p/.(n-'1+v..p) by A2,A5,A7,FINSEQ_6:174;
        hence W-bound L~q <= (Rotate(p,v)/.n)`1 by A1,A9,SPRECT_2:def 1;
        thus (Rotate(p,v)/.n)`1 <= E-bound L~q by A1,A10,A9,SPRECT_2:def 1;
        thus S-bound L~q <= (Rotate(p,v)/.n)`2 by A1,A10,A9,SPRECT_2:def 1;
        thus (Rotate(p,v)/.n)`2 <= N-bound L~q by A1,A10,A9,SPRECT_2:def 1;
      end;
      suppose
A11:    n > len(p:-v);
        then n > len p-v..p+1 by A2,FINSEQ_5:50;
        then n > 1+len p-v..p;
        then n+v..p > 1+len p by XREAL_1:19;
        then n+v..p-len p > 1 by XREAL_1:20;
        then
A12:    1 <= n+v..p-'len p by XREAL_0:def 2;
        v..p <= len p by A2,FINSEQ_4:21;
        then n+v..p <= len p+len p by A4,XREAL_1:7;
        then n+v..p-len p <= len p by XREAL_1:20;
        then n+v..p-'len p <= len p by XREAL_0:def 2;
        then n+v..p-'len p in Seg len p by A12,FINSEQ_1:1;
        then
A13:    n+v..p-'len p in dom p by FINSEQ_1:def 3;
A14:    Rotate(p,v)/.n = p/.(n+v..p-'len p) by A2,A4,A11,FINSEQ_6:177;
        hence W-bound L~q <= (Rotate(p,v)/.n)`1 by A1,A13,SPRECT_2:def 1;
        thus (Rotate(p,v)/.n)`1 <= E-bound L~q by A1,A14,A13,SPRECT_2:def 1;
        thus S-bound L~q <= (Rotate(p,v)/.n)`2 by A1,A14,A13,SPRECT_2:def 1;
        thus (Rotate(p,v)/.n)`2 <= N-bound L~q by A1,A14,A13,SPRECT_2:def 1;
      end;
    end;
    hence thesis by SPRECT_2:def 1;
  end;
  suppose
    not v in rng p;
    hence thesis by A1,FINSEQ_6:def 2;
  end;
end;
