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 Th72:
  L is being_line implies ex L0 st x in L0 & L0 // L
proof
  assume L is being_line;
  then consider y0,y1 such that
A1: y0 <> y1 and
A2: L = Line(y0,y1);
  set x9 = x + (y1 - y0);
  reconsider L0 = Line(x,x9) as Element of line_of_REAL n by Th47;
  take L0;
A3: y1 - y0 <> 0*n by A1,Th9;
A4: x9 - x = y1 - y0 by Th6;
  then x9 - x = 1 * (y1 - y0) by EUCLID_4:3;
  then (x9 - x) // (y1 - y0) by A4,A3;
  hence thesis by A2,EUCLID_4:9;
end;
