let V be non empty Abelian add-associative vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct ; for M1, M2, M3 being Subset of V
for z1, z2, z3 being Complex st M1 is convex & M2 is convex & M3 is convex holds
((z1 * M1) + (z2 * M2)) + (z3 * M3) is convex
let M1, M2, M3 be Subset of V; for z1, z2, z3 being Complex st M1 is convex & M2 is convex & M3 is convex holds
((z1 * M1) + (z2 * M2)) + (z3 * M3) is convex
let z1, z2, z3 be Complex; ( M1 is convex & M2 is convex & M3 is convex implies ((z1 * M1) + (z2 * M2)) + (z3 * M3) is convex )
assume that
A1:
( M1 is convex & M2 is convex )
and
A2:
M3 is convex
; ((z1 * M1) + (z2 * M2)) + (z3 * M3) is convex
(z1 * M1) + (z2 * M2) is convex
by A1, Th59;
then
(1r * ((z1 * M1) + (z2 * M2))) + (z3 * M3) is convex
by A2, Th59;
hence
((z1 * M1) + (z2 * M2)) + (z3 * M3) is convex
by Th51; verum