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;
reserve L,L0,L1,L2 for Element of line_of_REAL n;

theorem
  x in L & a<>1 & a*x in L implies 0*n in L
proof
  assume that
A1: x in L and
A2: a<>1 and
A3: a*x in L;
A4: 1-a<>0 by A2;
A5: 1-1/(1-a)+(1/(1-a))*a = 1-1/(1-a)+a/(1-a) by XCMPLX_1:99
    .= 1 + (-1/(1-a) + a/(1-a))
    .= 1 + ((-1)/(1-a) + a/(1-a)) by XCMPLX_1:187
    .= 1 + (-1+a)/(1-a) by XCMPLX_1:62
    .= 1 + (-(-a + 1))/(1-a)
    .= 1 + - (1-a)/(1-a) by XCMPLX_1:187
    .= 1 - (1-a)/(1-a)
    .= 1 - 1 by A4,XCMPLX_1:60
    .= 0;
  0*n = 0 * x by EUCLID_4:3
    .= (1-1/(1-a))*x +((1/(1-a))*a)*x by A5,EUCLID_4:7
    .= (1-1/(1-a))*x +(1/(1-a))*(a*x) by EUCLID_4:4;
  then
A6: 0*n in Line(x,a*x);
  Line(x,a*x) c= L by A1,A3,Th48;
  hence thesis by A6;
end;
