let V be RealLinearSpace; :: thesis: for p, q, r being Element of V st not p,q,r are_LinDep holds

for u, v being Element of V st not u is zero & not v is zero & not are_Prop u,v holds

ex y being Element of V st not u,v,y are_LinDep

let p, q, r be Element of V; :: thesis: ( not p,q,r are_LinDep implies for u, v being Element of V st not u is zero & not v is zero & not are_Prop u,v holds

ex y being Element of V st not u,v,y are_LinDep )

assume A1: not p,q,r are_LinDep ; :: thesis: for u, v being Element of V st not u is zero & not v is zero & not are_Prop u,v holds

ex y being Element of V st not u,v,y are_LinDep

let u, v be Element of V; :: thesis: ( not u is zero & not v is zero & not are_Prop u,v implies ex y being Element of V st not u,v,y are_LinDep )

assume A2: ( not u is zero & not v is zero & not are_Prop u,v ) ; :: thesis: not for y being Element of V holds u,v,y are_LinDep

assume A3: for y being Element of V holds u,v,y are_LinDep ; :: thesis: contradiction

then A4: u,v,r are_LinDep ;

( u,v,p are_LinDep & u,v,q are_LinDep ) by A3;

hence contradiction by A1, A2, A4, Th14; :: thesis: verum

for u, v being Element of V st not u is zero & not v is zero & not are_Prop u,v holds

ex y being Element of V st not u,v,y are_LinDep

let p, q, r be Element of V; :: thesis: ( not p,q,r are_LinDep implies for u, v being Element of V st not u is zero & not v is zero & not are_Prop u,v holds

ex y being Element of V st not u,v,y are_LinDep )

assume A1: not p,q,r are_LinDep ; :: thesis: for u, v being Element of V st not u is zero & not v is zero & not are_Prop u,v holds

ex y being Element of V st not u,v,y are_LinDep

let u, v be Element of V; :: thesis: ( not u is zero & not v is zero & not are_Prop u,v implies ex y being Element of V st not u,v,y are_LinDep )

assume A2: ( not u is zero & not v is zero & not are_Prop u,v ) ; :: thesis: not for y being Element of V holds u,v,y are_LinDep

assume A3: for y being Element of V holds u,v,y are_LinDep ; :: thesis: contradiction

then A4: u,v,r are_LinDep ;

( u,v,p are_LinDep & u,v,q are_LinDep ) by A3;

hence contradiction by A1, A2, A4, Th14; :: thesis: verum