let V be non empty addLoopStr ; :: thesis: for L being C_Linear_Combination of V
for v being Element of V holds
( L . v = 0c iff not v in Carrier L )
let L be C_Linear_Combination of V; :: thesis: for v being Element of V holds
( L . v = 0c iff not v in Carrier L )
let v be Element of V; :: thesis: ( L . v = 0c iff not v in Carrier L )
thus ( L . v = 0c implies not v in Carrier L ) :: thesis: ( not v in Carrier L implies L . v = 0c )

proof

assume A1: L . v = 0c ; :: thesis: not v in Carrier L
assume v in Carrier L ; :: thesis: contradiction
then ex u being Element of V st
( v = u & L . u <> 0c ) ;
hence contradiction by A1; :: thesis: verum

end;

thus ( not v in Carrier L implies L . v = 0c ) ; :: thesis: verum