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 Th77:
  x0 in L1 & x1 in L1 & x0 <> x1 & y0 in L2 & y1 in L2 & y0 <> y1
  & L1 // L2 implies x1 - x0 // y1 - y0
proof
  assume that
A1: x0 in L1 & x1 in L1 and
A2: x0 <> x1 and
A3: y0 in L2 & y1 in L2 and
A4: y0 <> y1 and
A5: L1 // L2;
  consider x2,x3,y2,y3 such that
A6: L1 = Line(x2,x3) and
A7: L2 = Line(y2,y3) and
A8: x3 - x2 // y3 - y2 by A5;
  consider t such that
A9: y1 - y0 = t*(y3 - y2) by A3,A7,Th31;
A10: x1 - x0 <> 0*n by A2,Th9;
A11: y1 - y0 <> 0*n by A4,Th9;
  then
A12: t <> 0 by A9,EUCLID_4:3;
  consider s such that
A13: x1 - x0 = s*(x3 - x2) by A1,A6,Th31;
  consider a such that
A14: x3 - x2 = a*(y3 - y2) by A8;
  x1 - x0 = (s*a)*(y3 - y2) by A13,A14,EUCLID_4:4
    .= (s*a)*(1 * (y3 - y2)) by EUCLID_4:3
    .= (s*a)*((1/t*t)*(y3 - y2)) by A12,XCMPLX_1:106
    .= (s*a)*(1/t*(t*(y3 - y2))) by EUCLID_4:4
    .= ((s*a)*(1/t))*(t*(y3 - y2)) by EUCLID_4:4
    .= ((s*a)/t)*(y1 - y0) by A9,XCMPLX_1:99;
  hence thesis by A10,A11;
end;
