let V be non empty addLoopStr ; :: thesis: for L being Linear_Combination of V

for v being Element of V holds

( L . v = 0 iff not v in Carrier L )

let L be 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 )

for v being Element of V holds

( L . v = 0 iff not v in Carrier L )

let L be 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

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