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

proof

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

end;

assume not v in Carrier L ; :: thesis: L . v = 0
hence L . v = 0 ; :: thesis: verum