
theorem
  for V being Abelian add-associative vector-distributive scalar-distributive
  scalar-associative scalar-unital non empty
  RLSStruct, M,N being Subset of V st M is convex & N is convex holds M + N is
  convex
proof
  let V be Abelian add-associative vector-distributive scalar-distributive
  scalar-associative scalar-unital non empty RLSStruct;
  let M,N be Subset of V;
  assume
A1: M is convex & N is convex;
  for u,v being VECTOR of V, r being Real
   st 0 < r & r < 1 & u in M+N & v
  in M+N holds r*u + (1-r)*v in M+N
  proof
    let u,v be VECTOR of V;
    let r be Real;
    assume that
A2: 0 < r & r < 1 and
A3: u in M+N and
A4: v in M+N;
    v in {x + y where x,y is Element of V : x in M & y in N} by A4,
RUSUB_4:def 9;
    then consider x2,y2 being Element of V such that
A5: v = x2 + y2 and
A6: x2 in M & y2 in N;
    u in {x + y where x,y is Element of V : x in M & y in N} by A3,
RUSUB_4:def 9;
    then consider x1,y1 being Element of V such that
A7: u = x1 + y1 and
A8: x1 in M & y1 in N;
A9: r*u + (1-r)*v = r*x1 + r*y1 + (1-r)*(x2+y2) by A7,A5,RLVECT_1:def 5
      .= r*x1 + r*y1 + ((1-r)*x2 + (1-r)*y2) by RLVECT_1:def 5
      .= r*x1 + r*y1 + (1-r)*x2 + (1-r)*y2 by RLVECT_1:def 3
      .= r*x1 + (1-r)*x2 + r*y1 + (1-r)*y2 by RLVECT_1:def 3
      .= (r*x1 + (1-r)*x2) + (r*y1 + (1-r)*y2) by RLVECT_1:def 3;
    r*x1 + (1-r)*x2 in M & r*y1 + (1-r)*y2 in N by A1,A2,A8,A6;
    then r*u + (1-r)*v in {x + y where x,y is Element of V : x in M & y in N}
    by A9;
    hence thesis by RUSUB_4:def 9;
  end;
  hence thesis;
end;
