reserve r,s,t,u for Real;

theorem Th19:
  for X being RealLinearSpace, M1,M2 being Subset of X st M1 c= M2
  holds Convex-Family M2 c= Convex-Family M1
proof
  let X be RealLinearSpace, M1,M2 be Subset of X such that
A1: M1 c= M2;
  let M be object;
  assume
A2: M in Convex-Family M2;
  then reconsider M as Subset of X;
  M2 c= M by A2,CONVEX1:def 4;
  then
A3: M1 c= M by A1;
  M is convex by A2,CONVEX1:def 4;
  hence thesis by A3,CONVEX1:def 4;
end;
