
theorem
  for V being RealLinearSpace, M being Subset of V, v being VECTOR of V
  holds M is convex iff v + M is convex
proof
  let V be RealLinearSpace;
  let M be Subset of V;
  let v be VECTOR of V;
A1: v + M is convex implies M is convex
  proof
    assume
A2: v + M is convex;
    let w1,w2 be VECTOR of V;
    let r be Real;
    assume that
A3: 0 < r & r < 1 and
A4: w1 in M and
A5: w2 in M;
    set x2 = v + w2;
    x2 in {v + w where w is VECTOR of V : w in M} by A5;
    then
A6: x2 in v + M by RUSUB_4:def 8;
    set x1 = v + w1;
A7: r*x1 + (1-r)*x2 = r*v + r*w1 + (1-r)*(v + w2) by RLVECT_1:def 5
      .= r*v + r*w1 + ((1-r)*v + (1-r)*w2) by RLVECT_1:def 5
      .= r*v + r*w1 + (1-r)*v + (1-r)*w2 by RLVECT_1:def 3
      .= r*v + (1-r)*v + r*w1 + (1-r)*w2 by RLVECT_1:def 3
      .= (r+(1-r))*v + r*w1 + (1-r)*w2 by RLVECT_1:def 6
      .= v + r*w1 + (1-r)*w2 by RLVECT_1:def 8
      .= v + (r*w1 + (1-r)*w2) by RLVECT_1:def 3;
    x1 in {v + w where w is VECTOR of V : w in M} by A4;
    then x1 in v + M by RUSUB_4:def 8;
    then r*x1 + (1-r)*x2 in v + M by A2,A3,A6;
    then v + (r*w1 + (1-r)*w2) in {v + w where w is VECTOR of V : w in M} by A7
,RUSUB_4:def 8;
    then ex w be VECTOR of V st v + (r*w1 + (1-r)*w2) = v + w & w in M;
    hence thesis by RLVECT_1:8;
  end;
  M is convex implies v + M is convex
  proof
    assume
A8: M is convex;
    let w1,w2 be VECTOR of V;
    let r be Real;
    assume that
A9: 0 < r & r < 1 and
A10: w1 in v + M and
A11: w2 in v + M;
    w2 in {v + w where w is VECTOR of V : w in M} by A11,RUSUB_4:def 8;
    then consider x2 be VECTOR of V such that
A12: w2 = v + x2 and
A13: x2 in M;
    w1 in {v + w where w is VECTOR of V : w in M} by A10,RUSUB_4:def 8;
    then consider x1 be VECTOR of V such that
A14: w1 = v + x1 and
A15: x1 in M;
A16: r*w1 + (1-r)*w2 = r*v + r*x1 + (1-r)*(v+x2) by A14,A12,RLVECT_1:def 5
      .= r*v + r*x1 + ((1-r)*v + (1-r)*x2) by RLVECT_1:def 5
      .= r*v + r*x1 + (1-r)*v + (1-r)*x2 by RLVECT_1:def 3
      .= r*v + (1-r)*v + r*x1 + (1-r)*x2 by RLVECT_1:def 3
      .= (r+(1-r))*v + r*x1 + (1-r)*x2 by RLVECT_1:def 6
      .= v + r*x1 + (1-r)*x2 by RLVECT_1:def 8
      .= v + (r*x1 + (1-r)*x2) by RLVECT_1:def 3;
    r*x1 + (1-r)*x2 in M by A8,A9,A15,A13;
    then r*w1 + (1-r)*w2 in {v + w where w is VECTOR of V : w in M} by A16;
    hence thesis by RUSUB_4:def 8;
  end;
  hence thesis by A1;
end;
