reserve a,a1,a2,a3,b,b1,b2,b3,r,s,t,u for Real;
reserve n for Nat;
reserve x0,x,x1,x2,x3,y0,y,y1,y2,y3 for Element of REAL n;

theorem Th40:
  x1 - x0, x3 - x2 are_lindependent2 & y0 in Line(x0,x1) & y1 in
  Line(x0,x1) & y0 <> y1 & y2 in Line(x2,x3) & y3 in Line(x2,x3) & y2 <> y3
  implies y1 - y0, y3 - y2 are_lindependent2
proof
  assume that
A1: x1 - x0, x3 - x2 are_lindependent2 and
A2: y0 in Line(x0,x1) & y1 in Line(x0,x1) and
A3: y0 <> y1 and
A4: y2 in Line(x2,x3) & y3 in Line(x2,x3) and
A5: y2 <> y3;
  consider s being Real such that
A6: y1 - y0 = s*(x1 - x0) by A2,Th31;
  consider t being Real such that
A7: y3 - y2 = t*(x3 - x2) by A4,Th31;
  for a,b st a*(y1 - y0)+b*(y3 - y2)=0*n holds a=0 & b=0
  proof
    let a,b;
    assume a*(y1 - y0)+b*(y3 - y2)=0*n;
    then
A8: 0*n = (a*s)*(x1 - x0)+b*(t*(x3 - x2)) by A6,A7,EUCLID_4:4
      .= (a*s)*(x1 - x0)+(b*t)*(x3 - x2) by EUCLID_4:4;
    then
A9: a*s= 0 by A1;
A10: b*t= 0 by A1,A8;
A11: b = b*t/t by A5,A7,Th10,XCMPLX_1:89
      .= 0 by A10;
    a = a*s/s by A3,A6,Th10,XCMPLX_1:89
      .= 0 by A9;
    hence thesis by A11;
  end;
  hence thesis;
end;
