
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 for r
  being Real holds r*M + (1-r)*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 that
A1: M is convex and
A2: N is convex;
  let r be Real;
  let u,v be VECTOR of V;
  let p be Real;
  assume that
A3: 0 < p & p < 1 and
A4: u in r*M + (1-r)*N and
A5: v in r*M + (1-r)*N;
A6: u in {x1 + y1 where x1,y1 is VECTOR of V : x1 in r*M & y1 in (1-r)*N} by A4
,RUSUB_4:def 9;
  v in {x2 + y2 where x2,y2 is VECTOR of V : x2 in r*M & y2 in (1-r)*N}
  by A5,RUSUB_4:def 9;
  then consider x2,y2 be VECTOR of V such that
A7: v = x2 + y2 and
A8: x2 in r*M and
A9: y2 in (1-r)*N;
  consider x1,y1 be VECTOR of V such that
A10: u = x1 + y1 and
A11: x1 in r*M and
A12: y1 in (1-r)*N by A6;
  consider mx2 be VECTOR of V such that
A13: x2 = r*mx2 and
A14: mx2 in M by A8;
  consider mx1 be VECTOR of V such that
A15: x1 = r*mx1 and
A16: mx1 in M by A11;
A17: p*x1 + (1-p)*x2 = p*r*mx1 + (1-p)*(r*mx2) by A15,A13,RLVECT_1:def 7
    .= p*r*mx1 + (1-p)*r*mx2 by RLVECT_1:def 7
    .= r*(p*mx1) + (1-p)*r*mx2 by RLVECT_1:def 7
    .= r*(p*mx1) + r*((1-p)*mx2) by RLVECT_1:def 7
    .= r*(p*mx1 + (1-p)*mx2) by RLVECT_1:def 5;
  consider ny2 be VECTOR of V such that
A18: y2 = (1-r)*ny2 and
A19: ny2 in N by A9;
  consider ny1 be VECTOR of V such that
A20: y1 = (1-r)*ny1 and
A21: ny1 in N by A12;
A22: p*y1 + (1-p)*y2 = p*(1-r)*ny1 + (1-p)*((1-r)*ny2) by A20,A18,
RLVECT_1:def 7
    .= p*(1-r)*ny1 + (1-p)*(1-r)*ny2 by RLVECT_1:def 7
    .= (1-r)*(p*ny1) + (1-p)*(1-r)*ny2 by RLVECT_1:def 7
    .= (1-r)*(p*ny1) + (1-r)*((1-p)*ny2) by RLVECT_1:def 7
    .= (1-r)*(p*ny1 + (1-p)*ny2) by RLVECT_1:def 5;
  p*ny1 + (1-p)*ny2 in N by A2,A3,A21,A19;
  then
A23: p*y1 + (1-p)*y2 in (1-r)*N by A22;
  p*mx1 + (1-p)*mx2 in M by A1,A3,A16,A14;
  then
A24: p*x1 + (1-p)*x2 in {r*w where w is VECTOR of V : w in M} by A17;
  p*u + (1-p)*v = p*x1 + p*y1 + (1-p)*(x2 + y2) by A10,A7,RLVECT_1:def 5
    .= p*x1 + p*y1 + ((1-p)*x2 + (1-p)*y2) by RLVECT_1:def 5
    .= p*x1 + p*y1 + (1-p)*x2 + (1-p)*y2 by RLVECT_1:def 3
    .= p*x1 + (1-p)*x2 + p*y1 + (1-p)*y2 by RLVECT_1:def 3
    .= (p*x1 + (1-p)*x2) + (p*y1 + (1-p)*y2) by RLVECT_1:def 3;
  then p*u + (1-p)*v in {w1 + w2 where w1,w2 is VECTOR of V : w1 in r*M & w2
  in (1-r)*N} by A24,A23;
  hence thesis by RUSUB_4:def 9;
end;
