reserve V for RealLinearSpace;
reserve p,q,r,u,v,w,y,u1,v1,w1 for Element of V;
reserve a,b,c,d,a1,b1,c1,a2,b2,c2,a3,b3,e,f for Real;

theorem Th14:
  not are_Prop p,q & p,q,u are_LinDep & p,q,v are_LinDep & p,q,w
  are_LinDep & p is not zero & q is not zero implies u,v,w are_LinDep
proof
  assume that
A1: not are_Prop p,q and
A2: p,q,u are_LinDep and
A3: p,q,v are_LinDep and
A4: p,q,w are_LinDep and
A5: p is not zero & q is not zero;
  consider a1,b1 such that
A6: u = a1*p + b1*q by A1,A2,A5,Th6;
  consider a3,b3 such that
A7: w = a3*p + b3*q by A1,A4,A5,Th6;
  consider a2,b2 such that
A8: v = a2*p + b2*q by A1,A3,A5,Th6;
  set a = a2*b3 - a3*b2, b = -(a1*b3) + a3*b1, c = a1*b2 - a2*b1;
A9: now
A10: w=0.V & v=0.V & (are_Prop v,w or v = 0.V or w = 0.V) implies thesis
    by Th10;
A11: w=0.V & u=0.V & (are_Prop v,w or v=0.V or w=0.V) implies thesis by Th10;
A12: u=0.V & v=0.V & (are_Prop v,w or v = 0.V or w = 0.V) implies thesis
    by Th10;
A13: ( w=0.V & are_Prop u,v & w=0.V or u=0.V & u=0.V & are_Prop v,w or
    are_Prop w,u & v=0.V & v=0.V ) implies thesis by Th11;
A14: are_Prop w,u & are_Prop u,v & are_Prop v,w implies thesis by Th11;
    assume that
A15: a=0 and
A16: b=0 and
A17: c =0;
    0 = a3*b1-a1*b3 by A16;
    hence thesis by A1,A5,A6,A8,A7,A15,A17,A14,A13,A11,A10,A12,Th9;
  end;
  0.V = (a*a1 + b*a2 + c*a3)*p & 0.V = (a*b1 + b*b2 + c*b3)*q by RLVECT_1:10;
  then
A18: 0.V = (a*a1 + b*a2 + c*a3)*p + (a*b1 + b*b2 + c*b3)*q;
  (a*a1 + b*a2 + c*a3)*p = (a*a1)*p + (b*a2)*p + (c*a3)*p by Lm3;
  then
  0.V = ((a*a1)*p + (b*a2)*p + (c*a3)*p) + ((a*b1)*q + (b*b2)*q + (c*b3)*
  q) by A18,Lm3;
  then
A19: 0.V = ((a*a1)*p+(a*b1)*q) + ((b*a2)*p+(b*b2)*q) + ((c*a3)*p+(c*b3)* q)
  by Lm4;
A20: ((c*a3)*p+(c*b3)*q) = c*(a3*p+b3*q) by Lm5;
  ( (a*a1)*p+(a*b1)*q) = a*(a1*p+b1*q) & ((b*a2)*p+(b*b2)*q) = b*(a2*p+b2
  *q) by Lm5;
  hence thesis by A6,A8,A7,A19,A20,A9;
end;
