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
  for A being Subset of REAL n,x st A is being_plane & x in A & ex a st
  a<>1 & a*x in A holds 0*n in A
proof
  let A be Subset of REAL n,x;
  assume that
A1: A is being_plane and
A2: x in A and
A3: ex a st a<>1 & a*x in A;
  consider a such that
A4: a<>1 and
A5: a*x in A by A3;
A6: 1-a<>0 by A4;
A7: 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 A6,XCMPLX_1:60
    .= 0;
  0*n = 0 * x by EUCLID_4:3
    .= (1-1/(1-a))*x +((1/(1-a))*a)*x by A7,EUCLID_4:7
    .= (1-1/(1-a))*x +(1/(1-a))*(a*x) by EUCLID_4:4;
  then
A8: 0*n in Line(x,a*x);
  ex x1,x2,x3 st x2-x1,x3-x1 are_lindependent2 & A = plane (x1,x2,x3) by A1;
  then Line(x,a*x) c= A by A2,A5,Th85;
  hence thesis by A8;
end;
