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 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 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 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 and
A2: v2 <> v3 and
A3: v3 <> v1 and
A4: L is convex ; :: thesis: ( ex r1, r2, r3 being Real 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) )
A5: Carrier L c= {v1,v2,v3} by Def4;
A6: Carrier L <> {} by A4, Th77;
ex r1, r2, r3 being Real 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 A5, A6, ZFMISC_1:118;

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

then not v2 in Carrier L by A1, TARSKI:def 1;
then A8: 0 = L . v2 ;
A9: (1 + 0) + 0 = 1 ;
not v3 in Carrier L by A3, A7, TARSKI:def 1;
then A10: 0 = L . v3 ;
ex r being Real st
( r = L . v1 & r = 1 ) by A4, A7, Th80;
hence ex r1, r2, r3 being Real st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) by A8, A10, A9; :: thesis: verum

end;

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

then not v1 in Carrier L by A1, TARSKI:def 1;
then A12: 0 = L . v1 ;
A13: (0 + 1) + 0 = 1 ;
not v3 in Carrier L by A2, A11, TARSKI:def 1;
then A14: 0 = L . v3 ;
ex r being Real st
( r = L . v2 & r = 1 ) by A4, A11, Th80;
hence ex r1, r2, r3 being Real st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) by A12, A14, A13; :: thesis: verum

end;

suppose A15: Carrier L = {v3} ; :: thesis: ex r1, r2, r3 being Real st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 )

then not v1 in Carrier L by A3, TARSKI:def 1;
then A16: 0 = L . v1 ;
A17: (0 + 0) + 1 = 1 ;
not v2 in Carrier L by A2, A15, TARSKI:def 1;
then A18: 0 = L . v2 ;
ex r being Real st
( r = L . v3 & r = 1 ) by A4, A15, Th80;
hence ex r1, r2, r3 being Real st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) by A16, A18, A17; :: thesis: verum

end;

suppose A19: Carrier L = {v1,v2} ; :: thesis: ex r1, r2, r3 being Real st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 )

set r3 = 0 ;
not v3 in { v where v is Element of V : L . v <> 0 } by A2, A3, A19, TARSKI:def 2;
then A20: 0 = L . v3 ;
consider r1, r2 being Real such that
A21: ( r1 = L . v1 & r2 = L . v2 ) and
A22: r1 + r2 = 1 and
A23: ( r1 >= 0 & r2 >= 0 ) by A1, A4, A19, Th81;
(r1 + r2) + 0 = 1 by A22;
hence ex r1, r2, r3 being Real st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) by A21, A23, A20; :: thesis: verum

end;

suppose A24: Carrier L = {v2,v3} ; :: thesis: ex r1, r2, r3 being Real st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 )

set r1 = 0 ;
not v1 in Carrier L by A1, A3, A24, TARSKI:def 2;
then A25: 0 = L . v1 ;
consider r2, r3 being Real such that
A26: ( r2 = L . v2 & r3 = L . v3 ) and
A27: r2 + r3 = 1 and
A28: ( r2 >= 0 & r3 >= 0 ) by A2, A4, A24, Th81;
(0 + r2) + r3 = 1 by A27;
hence ex r1, r2, r3 being Real st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) by A26, A28, A25; :: thesis: verum

end;

suppose A29: Carrier L = {v1,v3} ; :: thesis: ex r1, r2, r3 being Real st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 )

set r2 = 0 ;
not v2 in Carrier L by A1, A2, A29, TARSKI:def 2;
then A30: 0 = L . v2 ;
consider r1, r3 being Real such that
A31: ( r1 = L . v1 & r3 = L . v3 ) and
A32: r1 + r3 = 1 and
A33: ( r1 >= 0 & r3 >= 0 ) by A3, A4, A29, Th81;
(r1 + 0) + r3 = 1 by A32;
hence ex r1, r2, r3 being Real st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) by A31, A33, A30; :: thesis: verum

end;

suppose Carrier L = {v1,v2,v3} ; :: thesis: ex r1, r2, r3 being Real 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 st
( r1 = L . v1 & r2 = L . v2 & r3 = L . v3 & (r1 + r2) + r3 = 1 & r1 >= 0 & r2 >= 0 & r3 >= 0 ) by A1, A2, A3, A4, Th82; :: thesis: verum

end;

end;

end;

hence ( ex r1, r2, r3 being Real 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, A2, A3, Lm3; :: thesis: verum