let V be RealLinearSpace; :: thesis: for u, v being Element of V st ( for a, b being Real st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) ) holds
( not u is zero & not v is zero & not are_Prop u,v )
let u, v be Element of V; :: thesis: ( ( for a, b being Real st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) ) implies ( not u is zero & not v is zero & not are_Prop u,v ) )
assume A1: for a, b being Real st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) ; :: thesis: ( not u is zero & not v is zero & not are_Prop u,v )
thus ( not u is zero & not v is zero ) :: thesis: not are_Prop u,v

proof

A2: now

assume u is zero ; :: thesis: contradiction
then A3: u = 0. V by STRUCT_0:def 15;
0. V = (0. V) + (0. V) by RLVECT_1:10
.= (1 * u) + (0. V) by A3, RLVECT_1:def 9
.= (1 * u) + (0 * v) by RLVECT_1:23 ;
hence contradiction by A1; :: thesis: verum

end;

now

assume v is zero ; :: thesis: contradiction
then A4: v = 0. V by STRUCT_0:def 15;
0. V = (0. V) + (0. V) by RLVECT_1:10
.= (0. V) + (1 * v) by A4, RLVECT_1:def 9
.= (0 * u) + (1 * v) by RLVECT_1:23 ;
hence contradiction by A1; :: thesis: verum

end;

hence ( not u is zero & not v is zero ) by A2; :: thesis: verum

end;

given a, b being Real such that A5: a * u = b * v and
A6: ( a <> 0 & b <> 0 ) ; :: according to ANPROJ_1:def 2 :: thesis: contradiction
0. V = (a * u) - (b * v) by A5, RLVECT_1:28
.= (a * u) + (b * (- v)) by RLVECT_1:39
.= (a * u) + ((- b) * v) by RLVECT_1:38 ;
hence contradiction by A1, A6; :: thesis: verum