let V be ComplexLinearSpace; :: thesis: for v1, v2, v3 being VECTOR of V
for L being C_Linear_Combination of {v1,v2,v3} st v1 <> v2 & v2 <> v3 & v3 <> v1 & L is convex holds
( ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) & Sum L = (((L . v1) * v1) + ((L . v2) * v2)) + ((L . v3) * v3) )
let v1, v2, v3 be VECTOR of V; :: thesis: for L being C_Linear_Combination of {v1,v2,v3} st v1 <> v2 & v2 <> v3 & v3 <> v1 & L is convex holds
( ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) & Sum L = (((L . v1) * v1) + ((L . v2) * v2)) + ((L . v3) * v3) )
let L be C_Linear_Combination of {v1,v2,v3}; :: thesis: ( v1 <> v2 & v2 <> v3 & v3 <> v1 & L is convex implies ( ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) & Sum L = (((L . v1) * v1) + ((L . v2) * v2)) + ((L . v3) * v3) ) )
assume that
A1: ( v1 <> v2 & v2 <> v3 & v3 <> v1 ) and
A2: L is convex ; :: thesis: ( ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) & Sum L = (((L . v1) * v1) + ((L . v2) * v2)) + ((L . v3) * v3) )
A3: Carrier L c= {v1,v2,v3} by Def3;
A4: Carrier L <> {} by A2, Th221;
ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 )

proof

per cases ( Carrier L = {v1} or Carrier L = {v2} or Carrier L = {v3} or Carrier L = {v1,v2} or Carrier L = {v2,v3} or Carrier L = {v1,v3} or Carrier L = {v1,v2,v3} ) by A3, A4, ZFMISC_1:142;

suppose A5: Carrier L = {v1} ; :: thesis: ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 )

then A6: ex r being Real st
( r = L . v1 & r = 1 ) by A2, Lm211;
( not v2 in Carrier L & not v3 in Carrier L ) by A1, A5, TARSKI:def 1;
then ( 1 = L . v1 & 0 = L . v2 & 0 = L . v3 & (1 + 0 ) + 0 = 1 & 1 >= 0 & 0 >= 0 & 0 >= 0 ) by A6;
hence ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) ; :: thesis: verum

end;

suppose A7: Carrier L = {v2} ; :: thesis: ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 )

then A8: ex r being Real st
( r = L . v2 & r = 1 ) by A2, Lm211;
( not v1 in Carrier L & not v3 in Carrier L ) by A1, A7, TARSKI:def 1;
then ( 0 = L . v1 & 1 = L . v2 & 0 = L . v3 & (0 + 1) + 0 = 1 & 0 >= 0 & 1 >= 0 & 0 >= 0 ) by A8;
hence ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) ; :: thesis: verum

end;

suppose A9: Carrier L = {v3} ; :: thesis: ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 )

then A10: ex r being Real st
( r = L . v3 & r = 1 ) by A2, Lm211;
( not v1 in Carrier L & not v2 in Carrier L ) by A1, A9, TARSKI:def 1;
then ( 0 = L . v1 & 0 = L . v2 & 1 = L . v3 & (0 + 0 ) + 1 = 1 & 0 >= 0 & 0 >= 0 & 1 >= 0 ) by A10;
hence ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) ; :: thesis: verum

end;

suppose A11: Carrier L = {v1,v2} ; :: thesis: ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 )

then B1: not v3 in { v where v is Element of V : L . v <> 0 } by A1, TARSKI:def 2;
set r3 = 0 ;
consider r1, r2 being Real such that
A13: ( r1 = L . v1 & r2 = L . v2 & r1 + r2 = 1 & r1 >= 0 & r2 >= 0 ) by A1, A2, A11, Lm212;
( r1 = L . v1 & r2 = L . v2 & 0 = L . v3 & (r1 + r2) + 0 = 1 & r1 >= 0 & r2 >= 0 & 0 >= 0 ) by B1, A13;
hence ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) ; :: thesis: verum

end;

suppose A14: Carrier L = {v2,v3} ; :: thesis: ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 )

then B2: not v1 in Carrier L by A1, TARSKI:def 2;
set r1 = 0 ;
consider r2, r3 being Real such that
A16: ( r2 = L . v2 & r3 = L . v3 & r2 + r3 = 1 & r2 >= 0 & r3 >= 0 ) by A1, A2, A14, Lm212;
( 0 = L . v1 & r2 = L . v2 & r3 = L . v3 & (0 + r2) + r3 = 1 & 0 >= 0 & r2 >= 0 & r3 >= 0 ) by B2, A16;
hence ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) ; :: thesis: verum

end;

suppose A17: Carrier L = {v1,v3} ; :: thesis: ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 )

then B3: not v2 in Carrier L by A1, TARSKI:def 2;
set r2 = 0 ;
consider r1, r3 being Real such that
A19: ( r1 = L . v1 & r3 = L . v3 & r1 + r3 = 1 & r1 >= 0 & r3 >= 0 ) by A1, A2, A17, Lm212;
( r1 = L . v1 & 0 = L . v2 & r3 = L . v3 & (r1 + 0 ) + r3 = 1 & r1 >= 0 & 0 >= 0 & r3 >= 0 ) by B3, A19;
hence ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) ; :: thesis: verum

end;

suppose Carrier L = {v1,v2,v3} ; :: thesis: ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 )

hence ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) by A1, A2, Lm215; :: thesis: verum

end;

end;

end;

hence ( ex r1, r2, r3 being real number st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) & Sum L = (((L . v1) * v1) + ((L . v2) * v2)) + ((L . v3) * v3) ) by A1, Lm214; :: thesis: verum