reserve X for non empty BCIStr_1;
reserve d for Element of X;
reserve n,m,k for Nat;
reserve f for sequence of  the carrier of X;

theorem
  for X being BCI-Algebra_with_Condition(S), a,b being Element of
AtomSet(X) holds for ma being Element of X st (for x being Element of BranchV(a
  ) holds x <= ma) holds ex mb being Element of X st for y being Element of
  BranchV(b) holds y <= mb
proof
  let X be BCI-Algebra_with_Condition(S);
  let a,b be Element of AtomSet(X);
  let ma be Element of X;
  assume
A1: for x being Element of BranchV(a) holds x <= ma;
  ex mb being Element of X st for y being Element of BranchV(b) holds y <= mb
  proof
    set mb = (b * (0.X\a)) * ma;
    for y being Element of BranchV(b) holds y <= mb
    proof
      a\a = 0.X by BCIALG_1:def 5;
      then a <= a;
      then a in {yy2 where yy2 is Element of X:a <= yy2};
      then
A2:   a is Element of BranchV(a);
      let y be Element of BranchV(b);
      0.X in AtomSet(X);
      then consider x0 being Element of AtomSet(X) such that
A3:   x0 = 0.X;
      y in {yy where yy is Element of X:b<=yy};
      then ex yy being Element of X st y=yy & b<= yy;
      then b\b <= y\b by BCIALG_1:5;
      then y\b in {yy1 where yy1 is Element of X:b\b <= yy1};
      then
A4:   y\b is Element of BranchV(b\b);
A5:   (b\b)\(x0\a) = x0\(x0\a) by A3,BCIALG_1:def 5
        .= a by BCIALG_1:24;
      x0\x0 = 0.X by BCIALG_1:def 5;
      then x0 <= x0;
      then x0 in {yy2 where yy2 is Element of X:x0 <= yy2};
      then x0 is Element of BranchV(x0);
      then x0\a is Element of BranchV(x0\a) by A2,BCIALG_1:39;
      then (y\b)\(x0\a) is Element of BranchV(a) by A4,A5,BCIALG_1:39;
      then
A6:   (y\b)\(x0\a) <= ma by A1;
      y\mb = (y\(b * (0.X\a))) \ ma by Th11
        .= ((y\b)\(x0\a))\ma by A3,Th11
        .= 0.X by A6;
      hence thesis;
    end;
    hence thesis;
  end;
  hence thesis;
end;
