let V be RealLinearSpace; :: thesis: for u, v, w being Element of V st ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) holds
( not u is zero & not v is zero & not w is zero & not u,v,w are_LinDep & not are_Prop u,v )
let u, v, w be Element of V; :: thesis: ( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ) implies ( not u is zero & not v is zero & not w is zero & not u,v,w are_LinDep & not are_Prop u,v ) )
assume A1: for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds
( a = 0 & b = 0 & c = 0 ) ; :: thesis: ( not u is zero & not v is zero & not w is zero & not u,v,w are_LinDep & not are_Prop u,v )
thus ( not u is zero & not v is zero & not w is zero ) :: thesis: ( not u,v,w are_LinDep & 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
.= ((0. V) + (0. V)) + (0. V) by RLVECT_1:10
.= ((1 * u) + (0. V)) + (0. V) by A3, RLVECT_1:def 9
.= ((1 * u) + (0 * v)) + (0. V) by RLVECT_1:23
.= ((1 * u) + (0 * v)) + (0 * w) by RLVECT_1:23 ;
hence contradiction by A1; :: thesis: verum

end;

A4: now

assume v is zero ; :: thesis: contradiction
then A5: v = 0. V by STRUCT_0:def 15;
0. V = (0. V) + (0. V) by RLVECT_1:10
.= ((0. V) + (0. V)) + (0. V) by RLVECT_1:10
.= ((0. V) + (1 * v)) + (0. V) by A5, RLVECT_1:def 9
.= ((0 * u) + (1 * v)) + (0. V) by RLVECT_1:23
.= ((0 * u) + (1 * v)) + (0 * w) by RLVECT_1:23 ;
hence contradiction by A1; :: thesis: verum

end;

now

assume w is zero ; :: thesis: contradiction
then A6: w = 0. V by STRUCT_0:def 15;
0. V = (0. V) + (0. V) by RLVECT_1:10
.= ((0. V) + (0. V)) + (0. V) by RLVECT_1:10
.= ((0. V) + (0. V)) + (1 * w) by A6, RLVECT_1:def 9
.= ((0 * u) + (0. V)) + (1 * w) by RLVECT_1:23
.= ((0 * u) + (0 * v)) + (1 * w) by RLVECT_1:23 ;
hence contradiction by A1; :: thesis: verum

end;

hence ( not u is zero & not v is zero & not w is zero ) by A2, A4; :: thesis: verum

end;

thus not u,v,w are_LinDep by A1, ANPROJ_1:def 3; :: thesis: not are_Prop u,v
hence not are_Prop u,v by ANPROJ_1:17; :: thesis: verum