let V be non empty Abelian add-associative vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct ; :: thesis: for M, N being Subset of V st M is convex & N is convex holds
M + N is convex

let M, N be Subset of V; :: thesis: ( M is convex & N is convex implies M + N is convex )
assume A1: ( M is convex & N is convex ) ; :: thesis: M + N is convex
for u, v being VECTOR of V
for 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; :: thesis: for 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

let r be Real; :: thesis: ( 0 < r & r < 1 & u in M + N & v in M + N implies (r * u) + ((1 - r) * v) in M + N )
assume that
A2: ( 0 < r & r < 1 ) and
A3: u in M + N and
A4: v in M + N ; :: thesis: (r * u) + ((1 - r) * v) in M + N
v in { (x + y) where x, y is Element of V : ( x in M & y in N ) } by ;
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 ;
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
.= ((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 (r * u) + ((1 - r) * v) in M + N by RUSUB_4:def 9; :: thesis: verum
end;
hence M + N is convex ; :: thesis: verum