let V be non empty Abelian add-associative RealLinearSpace-like RLSStruct ; :: thesis: for M1, M2, M3 being Subset of V
for r1, r2, r3 being Real st M1 is convex & M2 is convex & M3 is convex holds
((r1 * M1) + (r2 * M2)) + (r3 * M3) is convex
let M1, M2, M3 be Subset of V; :: thesis: for r1, r2, r3 being Real st M1 is convex & M2 is convex & M3 is convex holds
((r1 * M1) + (r2 * M2)) + (r3 * M3) is convex
let r1, r2, r3 be Real; :: thesis: ( M1 is convex & M2 is convex & M3 is convex implies ((r1 * M1) + (r2 * M2)) + (r3 * M3) is convex )
assume that
A1: M1 is convex and
A2: M2 is convex and
A3: M3 is convex ; :: thesis: ((r1 * M1) + (r2 * M2)) + (r3 * M3) is convex
(r1 * M1) + (r2 * M2) is convex by A1, A2, Th11;
then (1 * ((r1 * M1) + (r2 * M2))) + (r3 * M3) is convex by A3, Th11;
hence ((r1 * M1) + (r2 * M2)) + (r3 * M3) is convex by Lm1; :: thesis: verum