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 Th11:
  (are_Prop u,v or are_Prop w,u or are_Prop v,w) implies w,u,v are_LinDep
proof
A1: for u,v,w st are_Prop u,v holds w,u,v are_LinDep
  proof
    let u,v,w;
A2: 0*w = 0.V by RLVECT_1:10;
    assume are_Prop u,v;
    then consider a,b such that
A3: a*u = b*v and
A4: a<>0 and
    b<>0;
    0.V=a*u + -b*v by A3,RLVECT_1:5
      .= a*u + (-1)*(b*v) by RLVECT_1:16
      .= a*u + (-1)*b*v by RLVECT_1:def 7;
    then 0.V=0*w + a*u + (-1)*b*v by A2;
    hence thesis by A4;
  end;
A5: now
    assume are_Prop w,u;
    then v,w,u are_LinDep by A1;
    hence thesis by Th5;
  end;
A6: now
    assume are_Prop v,w;
    then u,v,w are_LinDep by A1;
    hence thesis by Th5;
  end;
  assume are_Prop u,v or are_Prop w,u or are_Prop v,w;
  hence thesis by A1,A5,A6;
end;
