
theorem Th7:
  for V being non empty RLSStruct, M being cone Subset of V st V is
vector-distributive scalar-distributive scalar-associative scalar-unital
 holds M is convex iff for u,v being VECTOR of V st u in M
  & v in M holds u + v in M
proof
  let V be non empty RLSStruct;
  let M be cone Subset of V;
A1: (for u,v being VECTOR of V st u in M & v in M holds u + v in M) implies
  M is convex
  proof
    assume
A2: for u,v being VECTOR of V st u in M & v in M holds u + v in M;
    for u,v being VECTOR of V, r be Real
      st 0 < r & r < 1 & u in M & v in
    M holds r*u + (1-r)*v in M
    proof
      let u,v be VECTOR of V;
      let r be Real;
      assume that
A3:   0 < r and
A4:   r < 1 and
A5:   u in M and
A6:   v in M;
    reconsider r as Real;
      r + 0 < 1 by A4;
      then 1 - r > 0 by XREAL_1:20;
      then
A7:   (1-r)*v in M by A6,Def3;
      r*u in M by A3,A5,Def3;
      hence thesis by A2,A7;
    end;
    hence thesis by CONVEX1:def 2;
  end;
  assume
A8: V is vector-distributive scalar-distributive scalar-associative
scalar-unital;
  M is convex implies for u,v being VECTOR of V st u in M & v in M holds u
  + v in M
  proof
    assume
A9: M is convex;
    for u,v being VECTOR of V st u in M & v in M holds u + v in M
    proof
      let u,v being VECTOR of V;
      assume u in M & v in M;
      then (1/2)*u + (1-(1/2))*v in M by A9,CONVEX1:def 2;
      then
A10:  2*(jd*u + jd*v) in M by Def3;
      2*((1/2)*u + (1/2)*v) = 2*((1/2)*u) + 2*((1/2)*v) by A8,RLVECT_1:def 5
        .= (2*(1/2))*u + 2*((1/2)*v) by A8,RLVECT_1:def 7
        .= 1*u + (2*(1/2))*v by A8,RLVECT_1:def 7
        .= u + 1*v by A8,RLVECT_1:def 8;
      hence thesis by A8,A10,RLVECT_1:def 8;
    end;
    hence thesis;
  end;
  hence thesis by A1;
end;
