
theorem
  for V being RealUnitarySpace-like non empty UNITSTR, M being Subset
  of V, v being VECTOR of V, r being Real
  st M = {u where u is VECTOR of V : u .|. v <= r} holds M is convex
proof
  let V be RealUnitarySpace-like non empty UNITSTR;
  let M be Subset of V;
  let v be VECTOR of V;
  let r be Real;
  assume
A1: M = {u where u is VECTOR of V : u.|.v <= r};
  let x,y be VECTOR of V;
  let p be Real;
  assume that
A2: 0 < p and
A3: p < 1 and
A4: x in M and
A5: y in M;
  0 + p < 1 by A3;
  then
A6: 0 < 1-p by XREAL_1:20;
  consider u2 be VECTOR of V such that
A7: y = u2 and
A8: u2.|.v <= r by A1,A5;
  ((1-p)*y).|.v = (1-p)*(u2.|.v) by A7,BHSP_1:def 2;
  then
A9: ((1-p)*y).|.v <= (1-p)*r by A8,A6,XREAL_1:64;
  consider u1 be VECTOR of V such that
A10: x = u1 and
A11: u1.|.v <= r by A1,A4;
  (p*x).|.v = p*(u1.|.v) by A10,BHSP_1:def 2;
  then
A12: (p*x).|.v <= p*r by A2,A11,XREAL_1:64;
  (p*x + (1-p)*y).|.v = (p*x).|.v + ((1-p)*y).|.v by BHSP_1:def 2;
  then (p*x + (1-p)*y).|.v <= p*r + (1-p)*r by A12,A9,XREAL_1:7;
  hence thesis by A1;
end;
