let V be RealLinearSpace; for u, v, u1, v1 being Element of V st ( for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) holds
( not u is zero & not v is zero & not are_Prop u,v & not u1 is zero & not v1 is zero & not are_Prop u1,v1 & not u,v,u1 are_LinDep & not u1,v1,u are_LinDep )
let u, v, u1, v1 be Element of V; ( ( for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) implies ( not u is zero & not v is zero & not are_Prop u,v & not u1 is zero & not v1 is zero & not are_Prop u1,v1 & not u,v,u1 are_LinDep & not u1,v1,u are_LinDep ) )
assume A1:
for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds
( a = 0 & b = 0 & a1 = 0 & b1 = 0 )
; ( not u is zero & not v is zero & not are_Prop u,v & not u1 is zero & not v1 is zero & not are_Prop u1,v1 & not u,v,u1 are_LinDep & not u1,v1,u are_LinDep )
A2:
now for d1, d2, d3 being Real st ((d1 * u) + (d2 * v)) + (d3 * u1) = 0. V holds
( d1 = 0 & d2 = 0 & d3 = 0 )end;
now for d1, d2, d3 being Real st ((d1 * u1) + (d2 * v1)) + (d3 * u) = 0. V holds
( d1 = 0 & d2 = 0 & d3 = 0 )end;
hence
( not u is zero & not v is zero & not are_Prop u,v & not u1 is zero & not v1 is zero & not are_Prop u1,v1 & not u,v,u1 are_LinDep & not u1,v1,u are_LinDep )
by A2, Th1; verum