reserve z,z1,z2 for Complex;
reserve r,x1,x2 for Real;
reserve p0,p,p1,p2,p3,q for Point of TOP-REAL 2;

theorem Th50:
  for n being Element of NAT,q1,q2 being Point of TOP-REAL n st q1
  ,q2 are_lindependent2 holds q1<>q2 & q1<>0.TOP-REAL n & q2<>0.TOP-REAL n
proof
  let n be Element of NAT,q1,q2 be Point of TOP-REAL n;
  assume
A1: q1,q2 are_lindependent2;
  assume
A2: q1=q2 or q1=0.TOP-REAL n or q2=0.TOP-REAL n;
  now
    per cases by A2;
    case
A3:   q1=q2;
      1*q1+(-1)*q2=1*q1+(-q2) by RLVECT_1:16
        .= q1+(-q2) by RLVECT_1:def 8
        .=0.TOP-REAL n by A3,RLVECT_1:5;
      hence contradiction by A1;
    end;
    case
      q1=0.TOP-REAL n;
      then 1*q1+(0)*q2=0.TOP-REAL n+(0)*q2 by RLVECT_1:10
        .=0.TOP-REAL n+0.TOP-REAL n by RLVECT_1:10
        .=0.TOP-REAL n by RLVECT_1:4;
      hence contradiction by A1;
    end;
    case
      q2=0.TOP-REAL n;
      then (0)*q1+1*q2=(0)*q1+0.TOP-REAL n by RLVECT_1:10
        .=0.TOP-REAL n+0.TOP-REAL n by RLVECT_1:10
        .=0.TOP-REAL n by RLVECT_1:4;
      hence contradiction by A1;
    end;
  end;
  hence contradiction;
end;
