
theorem
  for V being RealUnitarySpace-like non empty UNITSTR, M being Subset
of V, F being FinSequence of the carrier of V, B being FinSequence of REAL st M
= {u where u is VECTOR of V : for i being Element of NAT st i in (dom F /\ dom
B) holds ex v being VECTOR of V st v = F.i & u .|. v <= B.i} holds M is convex
proof
  let V be RealUnitarySpace-like non empty UNITSTR;
  let M be Subset of V;
  let F be FinSequence of the carrier of V;
  let B be FinSequence of REAL;
  assume
A1: M = {u where u is VECTOR of V: for i being Element of NAT st i in (
  dom F /\ dom B) holds ex v being VECTOR of V st v = F.i & u.|.v <= B.i};
  let u1,v1 be VECTOR of V;
  let r be Real;
  assume that
A2: 0 < r and
A3: r < 1 and
A4: u1 in M and
A5: v1 in M;
  consider v9 be VECTOR of V such that
A6: v1 = v9 and
A7: for i being Element of NAT st i in (dom F /\ dom B) holds ex v being
  VECTOR of V st v = F.i & v9.|.v <= B.i by A1,A5;
  consider u9 be VECTOR of V such that
A8: u1 = u9 and
A9: for i being Element of NAT st i in (dom F /\ dom B) holds ex v being
  VECTOR of V st v = F.i & u9.|.v <= B.i by A1,A4;
  for i being Element of NAT st i in (dom F /\ dom B) holds ex v being
  VECTOR of V st v = F.i & (r*u1 + (1-r)*v1).|.v <= B.i
  proof
    0 + r < 1 by A3;
    then
A10: 1 - r > 0 by XREAL_1:20;
    let i be Element of NAT;
    assume
A11: i in dom F /\ dom B;
    reconsider b = B.i as Real;
    consider x being VECTOR of V such that
A12: x = F.i and
A13: u9.|. x <= b by A9,A11;
    take v = x;
A14: (r*u1 + (1-r)*v1).|.v = (r*u1).|.v + ((1-r)*v1).|.v by BHSP_1:def 2
      .= r*(u1.|.v) + ((1-r)*v1).|.v by BHSP_1:def 2
      .= r*(u1.|.v) + (1-r)*(v1.|.v) by BHSP_1:def 2;
    r*(u1.|.v) <= r*b by A2,A8,A13,XREAL_1:64;
    then (r*u1 + (1-r)*v1).|.v <= r * b + (1-r)*(v1.|.v) by A14,XREAL_1:6;
    then
A15: (r*u1 + (1-r)*v1).|.v - r * b <= (1-r)*(v1.|.v) by XREAL_1:20;
    ex y being VECTOR of V st y = F.i & v9.|. y <= b by A7,A11;
    then (1-r)*(v1.|.v) <= (1-r)*b by A6,A12,A10,XREAL_1:64;
    then (r*u1 + (1-r)*v1).|.v - r * b <= (1-r)*b by A15,XXREAL_0:2;
    then (r*u1 + (1-r)*v1).|.v <= r * b + (1-r)*b by XREAL_1:20;
    hence thesis by A12;
  end;
  hence thesis by A1;
end;
