reserve i,j,k,n,m for Nat;
reserve p,q for Point of TOP-REAL 2;
reserve G for Go-board;
reserve C for Subset of TOP-REAL 2;

theorem Th42:
  for f being FinSequence of TOP-REAL 2, p,q being Point of
TOP-REAL 2 holds <*p,q*> is_in_the_area_of f iff <*p*> is_in_the_area_of f & <*
  q*> is_in_the_area_of f
proof
  let f be FinSequence of TOP-REAL 2, p,q be Point of TOP-REAL 2;
  thus <*p,q*> is_in_the_area_of f implies <*p*> is_in_the_area_of f & <*q*>
  is_in_the_area_of f
  proof
A1: dom<*p,q*> = {1,2} by FINSEQ_1:2,89;
    then
A2: 1 in dom<*p,q*> by TARSKI:def 2;
    assume
A3: <*p,q*> is_in_the_area_of f;
A4: <*p,q*>/.1 = p by FINSEQ_4:17;
    then
A5: p`1 <= E-bound L~f by A3,A2;
A6: p`2 <= N-bound L~f by A3,A2,A4;
A7: S-bound L~f <= p`2 by A3,A2,A4;
A8: W-bound L~f <= p`1 by A3,A2,A4;
    thus <*p*> is_in_the_area_of f
    proof
      let i;
      assume i in dom<*p*>;
      then i in {1} by FINSEQ_1:2,38;
      then i = 1 by TARSKI:def 1;
      hence thesis by A8,A5,A7,A6,FINSEQ_4:16;
    end;
    let i;
    assume i in dom<*q*>;
    then i in {1} by FINSEQ_1:2,38;
    then
A9: i = 1 by TARSKI:def 1;
A10: <*p,q*>/.2 = q by FINSEQ_4:17;
A11: 2 in dom<*p,q*> by A1,TARSKI:def 2;
    then
A12: q`1 <= E-bound L~f by A3,A10;
A13: q`2 <= N-bound L~f by A3,A11,A10;
A14: S-bound L~f <= q`2 by A3,A11,A10;
    W-bound L~f <= q`1 by A3,A11,A10;
    hence thesis by A12,A14,A13,A9,FINSEQ_4:16;
  end;
A15: <*p*>/.1 = p by FINSEQ_4:16;
  dom<*q*> = {1} by FINSEQ_1:2,38;
  then
A16: 1 in dom<*q*> by TARSKI:def 1;
  dom<*p*> = {1} by FINSEQ_1:2,38;
  then
A17: 1 in dom<*p*> by TARSKI:def 1;
  assume
A18: <*p*> is_in_the_area_of f;
  then
A19: p`1 <= E-bound L~f by A17,A15;
A20: p`2 <= N-bound L~f by A18,A17,A15;
A21: S-bound L~f <= p`2 by A18,A17,A15;
A22: <*q*>/.1 = q by FINSEQ_4:16;
  assume
A23: <*q*> is_in_the_area_of f;
  then
A24: W-bound L~f <= q`1 by A16,A22;
A25: q`1 <= E-bound L~f by A23,A16,A22;
  let i;
  assume i in dom<*p,q*>;
  then i in {1,2} by FINSEQ_1:2,89;
  then
A26: i =1 or i =2 by TARSKI:def 2;
A27: q`2 <= N-bound L~f by A23,A16,A22;
A28: S-bound L~f <= q`2 by A23,A16,A22;
  W-bound L~f <= p`1 by A18,A17,A15;
  hence thesis by A19,A21,A20,A24,A25,A28,A27,A26,FINSEQ_4:17;
end;
