let V be non empty Abelian add-associative vector-distributive scalar-distributive scalar-associative scalar-unital CLSStruct ; for M1, M2 being Subset of V
for z1, z2 being Complex st M1 is convex & M2 is convex holds
(z1 * M1) + (z2 * M2) is convex
let M1, M2 be Subset of V; for z1, z2 being Complex st M1 is convex & M2 is convex holds
(z1 * M1) + (z2 * M2) is convex
let z1, z2 be Complex; ( M1 is convex & M2 is convex implies (z1 * M1) + (z2 * M2) is convex )
assume that
A1:
M1 is convex
and
A2:
M2 is convex
; (z1 * M1) + (z2 * M2) is convex
let u, v be VECTOR of V; CONVEX4:def 23 for z being Complex st ex r being Real st
( z = r & 0 < r & r < 1 ) & u in (z1 * M1) + (z2 * M2) & v in (z1 * M1) + (z2 * M2) holds
(z * u) + ((1r - z) * v) in (z1 * M1) + (z2 * M2)
let s be Complex; ( ex r being Real st
( s = r & 0 < r & r < 1 ) & u in (z1 * M1) + (z2 * M2) & v in (z1 * M1) + (z2 * M2) implies (s * u) + ((1r - s) * v) in (z1 * M1) + (z2 * M2) )
assume that
A3:
ex p being Real st
( s = p & 0 < p & p < 1 )
and
A4:
u in (z1 * M1) + (z2 * M2)
and
A5:
v in (z1 * M1) + (z2 * M2)
; (s * u) + ((1r - s) * v) in (z1 * M1) + (z2 * M2)
consider v1, v2 being VECTOR of V such that
A6:
v = v1 + v2
and
A7:
v1 in z1 * M1
and
A8:
v2 in z2 * M2
by A5;
consider u1, u2 being VECTOR of V such that
A9:
u = u1 + u2
and
A10:
u1 in z1 * M1
and
A11:
u2 in z2 * M2
by A4;
consider y1 being VECTOR of V such that
A12:
v1 = z1 * y1
and
A13:
y1 in M1
by A7;
consider x1 being VECTOR of V such that
A14:
u1 = z1 * x1
and
A15:
x1 in M1
by A10;
A16: (s * u1) + ((1r - s) * v1) =
((z1 * s) * x1) + ((1r - s) * (z1 * y1))
by A14, A12, CLVECT_1:def 4
.=
((z1 * s) * x1) + ((z1 * (1r - s)) * y1)
by CLVECT_1:def 4
.=
(z1 * (s * x1)) + ((z1 * (1r - s)) * y1)
by CLVECT_1:def 4
.=
(z1 * (s * x1)) + (z1 * ((1r - s) * y1))
by CLVECT_1:def 4
.=
z1 * ((s * x1) + ((1r - s) * y1))
by CLVECT_1:def 2
;
consider y2 being VECTOR of V such that
A17:
v2 = z2 * y2
and
A18:
y2 in M2
by A8;
consider x2 being VECTOR of V such that
A19:
u2 = z2 * x2
and
A20:
x2 in M2
by A11;
A21: (s * u2) + ((1r - s) * v2) =
((z2 * s) * x2) + ((1r - s) * (z2 * y2))
by A19, A17, CLVECT_1:def 4
.=
((z2 * s) * x2) + ((z2 * (1r - s)) * y2)
by CLVECT_1:def 4
.=
(z2 * (s * x2)) + ((z2 * (1r - s)) * y2)
by CLVECT_1:def 4
.=
(z2 * (s * x2)) + (z2 * ((1r - s) * y2))
by CLVECT_1:def 4
.=
z2 * ((s * x2) + ((1r - s) * y2))
by CLVECT_1:def 2
;
(s * x2) + ((1r - s) * y2) in M2
by A2, A3, A20, A18;
then A22:
(s * u2) + ((1r - s) * v2) in z2 * M2
by A21;
(s * x1) + ((1r - s) * y1) in M1
by A1, A3, A15, A13;
then A23:
(s * u1) + ((1r - s) * v1) in z1 * M1
by A16;
(s * (u1 + u2)) + ((1r - s) * (v1 + v2)) =
((s * u1) + (s * u2)) + ((1r - s) * (v1 + v2))
by CLVECT_1:def 2
.=
((s * u1) + (s * u2)) + (((1r - s) * v1) + ((1r - s) * v2))
by CLVECT_1:def 2
.=
(((s * u1) + (s * u2)) + ((1r - s) * v1)) + ((1r - s) * v2)
by RLVECT_1:def 3
.=
(((s * u1) + ((1r - s) * v1)) + (s * u2)) + ((1r - s) * v2)
by RLVECT_1:def 3
.=
((s * u1) + ((1r - s) * v1)) + ((s * u2) + ((1r - s) * v2))
by RLVECT_1:def 3
;
hence
(s * u) + ((1r - s) * v) in (z1 * M1) + (z2 * M2)
by A9, A6, A23, A22; verum