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 Th44:
  for f being FinSequence of TOP-REAL 2, p,q being Point of
  TOP-REAL 2, r being Real
    st 0<=r & r <= 1 & <*p,q*> is_in_the_area_of f holds
  <*(1-r)*p+r*q*> is_in_the_area_of f
proof
  let f be FinSequence of TOP-REAL 2, p,q be Point of TOP-REAL 2,
      r be Real
  such that
A1: 0 <= r and
A2: r <= 1 and
A3: <*p,q*> is_in_the_area_of f;
A4: dom<*p,q*> = {1,2} by FINSEQ_1:2,89;
  then
A5: 2 in dom<*p,q*> by TARSKI:def 2;
A6: <*p,q*>/.2 = q by FINSEQ_4:17;
  then W-bound L~f <= q`1 by A3,A5;
  then
A7: r*W-bound L~f <= r*q`1 by A1,XREAL_1:64;
  q`1 <= E-bound L~f by A3,A5,A6;
  then
A8: r*q`1<= r*E-bound L~f by A1,XREAL_1:64;
A9: <*p,q*>/.1 = p by FINSEQ_4:17;
A10: 1-r >= 0 by A2,XREAL_1:48;
A11: 1 in dom<*p,q*> by A4,TARSKI:def 2;
  then W-bound L~f <= p`1 by A3,A9;
  then
A12: (1-r)*W-bound L~f <= (1-r)*p`1 by A10,XREAL_1:64;
  p`1 <= E-bound L~f by A3,A11,A9;
  then
A13: (1-r)*p`1 <= (1-r)*E-bound L~f by A10,XREAL_1:64;
  S-bound L~f <= p`2 by A3,A11,A9;
  then
A14: (1-r)*S-bound L~f <= (1-r)*p`2 by A10,XREAL_1:64;
A15: ((1-r)*p+r*q)`1 = ((1-r)*p)`1+(r*q)`1 by TOPREAL3:2
    .= (1-r)*p`1+(r*q)`1 by TOPREAL3:4
    .= (1-r)*p`1+r*q`1 by TOPREAL3:4;
  p`2 <= N-bound L~f by A3,A11,A9;
  then
A16: (1-r)*p`2 <= (1-r)*N-bound L~f by A10,XREAL_1:64;
  let n;
A17: dom <*(1-r)*p+r*q*> = {1} by FINSEQ_1:2,38;
  assume n in dom <*(1-r)*p+r*q*>;
  then
A18: n = 1 by A17,TARSKI:def 1;
  (1-r)*W-bound L~f+r*W-bound L~f = 1*W-bound L~f;
  then W-bound L~f <= (1-r)*p`1+r*q`1 by A7,A12,XREAL_1:7;
  hence W-bound L~f <= (<*(1-r)*p+r*q*>/.n)`1 by A18,A15,FINSEQ_4:16;
  (1-r)*E-bound L~f+r*E-bound L~f = 1*E-bound L~f;
  then (1-r)*p`1+r*q`1 <= E-bound L~f by A8,A13,XREAL_1:7;
  hence (<*(1-r)*p+r*q*>/.n)`1 <= E-bound L~f by A18,A15,FINSEQ_4:16;
A19: ((1-r)*p+r*q)`2 = ((1-r)*p)`2+(r*q)`2 by TOPREAL3:2
    .= (1-r)*p`2+(r*q)`2 by TOPREAL3:4
    .= (1-r)*p`2+r*q`2 by TOPREAL3:4;
  q`2 <= N-bound L~f by A3,A5,A6;
  then
A20: r*q`2<= r*N-bound L~f by A1,XREAL_1:64;
  S-bound L~f <= q`2 by A3,A5,A6;
  then
A21: r*S-bound L~f <= r*q`2 by A1,XREAL_1:64;
  (1-r)*S-bound L~f+r*S-bound L~f = 1*S-bound L~f;
  then S-bound L~f <= (1-r)*p`2+r*q`2 by A21,A14,XREAL_1:7;
  hence S-bound L~f <= (<*(1-r)*p+r*q*>/.n)`2 by A18,A19,FINSEQ_4:16;
  (1-r)*N-bound L~f+r*N-bound L~f = 1*N-bound L~f;
  then (1-r)*p`2+r*q`2 <= N-bound L~f by A20,A16,XREAL_1:7;
  hence thesis by A18,A19,FINSEQ_4:16;
end;
