let V be ComplexLinearSpace; :: thesis: for v1, v2 being VECTOR of V
for L being C_Linear_Combination of {v1,v2} st v1 <> v2 & L is convex holds
( ex r1, r2 being real number st
( r1 = L . v1 & r2 = L . v2 & r1 >= 0 & r2 >= 0 ) & Sum L = ((L . v1) * v1) + ((L . v2) * v2) )
let v1, v2 be VECTOR of V; :: thesis: for L being C_Linear_Combination of {v1,v2} st v1 <> v2 & L is convex holds
( ex r1, r2 being real number st
( r1 = L . v1 & r2 = L . v2 & r1 >= 0 & r2 >= 0 ) & Sum L = ((L . v1) * v1) + ((L . v2) * v2) )
let L be C_Linear_Combination of {v1,v2}; :: thesis: ( v1 <> v2 & L is convex implies ( ex r1, r2 being real number st
( r1 = L . v1 & r2 = L . v2 & r1 >= 0 & r2 >= 0 ) & Sum L = ((L . v1) * v1) + ((L . v2) * v2) ) )
assume that
A1: v1 <> v2 and
A2: L is convex ; :: thesis: ( ex r1, r2 being real number st
( r1 = L . v1 & r2 = L . v2 & r1 >= 0 & r2 >= 0 ) & Sum L = ((L . v1) * v1) + ((L . v2) * v2) )
A3: Carrier L c= {v1,v2} by Def3;
A4: Carrier L <> {} by A2, Th221;
ex r1, r2 being real number st
( r1 = L . v1 & r2 = L . v2 & r1 >= 0 & r2 >= 0 )

proof

per cases ( Carrier L = {v1} or Carrier L = {v2} or Carrier L = {v1,v2} ) by A3, A4, ZFMISC_1:42;

suppose A5: Carrier L = {v1} ; :: thesis: ex r1, r2 being real number st
( r1 = L . v1 & r2 = L . v2 & r1 >= 0 & r2 >= 0 )

then A6: ( ex r being Real st
( r = L . v1 & r = 1 ) & Sum L = (L . v1) * v1 ) by A2, Lm211;
not v2 in Carrier L by A1, A5, TARSKI:def 1;
then ( 1 = L . v1 & 0 = L . v2 & 1 + 0 = 1 & 1 >= 0 & 0 >= 0 ) by A6;
hence ex r1, r2 being real number st
( r1 = L . v1 & r2 = L . v2 & r1 >= 0 & r2 >= 0 ) ; :: thesis: verum

end;

suppose A7: Carrier L = {v2} ; :: thesis: ex r1, r2 being real number st
( r1 = L . v1 & r2 = L . v2 & r1 >= 0 & r2 >= 0 )

then A8: ( ex r being Real st
( r = L . v2 & r = 1 ) & Sum L = (L . v2) * v2 ) by A2, Lm211;
not v1 in Carrier L by A1, A7, TARSKI:def 1;
then ( 0 = L . v1 & 1 = L . v2 & 0 + 1 = 1 & 0 >= 0 & 1 >= 0 ) by A8;
hence ex r1, r2 being real number st
( r1 = L . v1 & r2 = L . v2 & r1 >= 0 & r2 >= 0 ) ; :: thesis: verum

end;

suppose Carrier L = {v1,v2} ; :: thesis: ex r1, r2 being real number st
( r1 = L . v1 & r2 = L . v2 & r1 >= 0 & r2 >= 0 )

then ex r1, r2 being Real st
( r1 = L . v1 & r2 = L . v2 & r1 + r2 = 1 & r1 >= 0 & r2 >= 0 ) by A1, A2, Lm212;
hence ex r1, r2 being real number st
( r1 = L . v1 & r2 = L . v2 & r1 >= 0 & r2 >= 0 ) ; :: thesis: verum

end;

end;

end;

hence ( ex r1, r2 being real number st
( r1 = L . v1 & r2 = L . v2 & r1 >= 0 & r2 >= 0 ) & Sum L = ((L . v1) * v1) + ((L . v2) * v2) ) by A1, Th51; :: thesis: verum