let V be RealLinearSpace; :: thesis: for L being Linear_Combination of V
for v being VECTOR of V st L is convex & L . v <= 0 holds
not v in Carrier L

let L be Linear_Combination of V; :: thesis: for v being VECTOR of V st L is convex & L . v <= 0 holds
not v in Carrier L

let v be VECTOR of V; :: thesis: ( L is convex & L . v <= 0 implies not v in Carrier L )
assume that
A1: L is convex and
A2: L . v <= 0 ; :: thesis: not v in Carrier L
per cases ( L . v = 0 or L . v < 0 ) by A2;
suppose L . v = 0 ; :: thesis: not v in Carrier L
end;
suppose A3: L . v < 0 ; :: thesis: not v in Carrier L
now :: thesis: not v in Carrier L
consider F being FinSequence of the carrier of V such that
F is one-to-one and
A4: rng F = Carrier L and
A5: ex f being FinSequence of REAL st
( len f = len F & Sum f = 1 & ( for n being Nat st n in dom f holds
( f . n = L . (F . n) & f . n >= 0 ) ) ) by A1;
assume v in Carrier L ; :: thesis: contradiction
then consider n being object such that
A6: n in dom F and
A7: F . n = v by ;
reconsider n = n as Element of NAT by A6;
consider f being FinSequence of REAL such that
A8: len f = len F and
Sum f = 1 and
A9: for n being Nat st n in dom f holds
( f . n = L . (F . n) & f . n >= 0 ) by A5;
n in Seg (len F) by ;
then A10: n in dom f by ;
then L . v = f . n by A9, A7;
hence contradiction by A3, A9, A10; :: thesis: verum
end;
hence not v in Carrier L ; :: thesis: verum
end;
end;