reserve r,s,t,u for Real;
reserve V for RealLinearSpace,
  v,w for Point of V;
reserve x1,x2,x3,x4,y1,y2 for Element of V;

theorem Th75:
  y1 in Line(x1,x2) & y2 in Line(x1,x2) & y1 <> y2 implies
    Line(y1,y2) = Line(x1,x2)
proof
  assume
A1: y1 in Line(x1,x2);
  then consider t such that
A2: y1 = (1-t)*x1 + t*x2;
  assume
A3: y2 in Line(x1,x2);
  then consider s such that
A4: y2 = (1-s)*x1 + s*x2;
  assume y1<>y2;
  then
A5: t-s<>0 by A2,A4;
  thus Line(y1,y2) c= Line(x1,x2) by A1,A3,Th74;
  (1-((t-1)/(t-s)))*y1 + ((t-1)/(t-s))*y2 = ((1*(t-s)-(t-1))/(t-s))*y1 + (
  (t-1)/(t-s))*y2 by A5,XCMPLX_1:127
    .= (((-s+1)/(t-s))*((1-t)*x1) + ((-s+1)/(t-s))*(t*x2)) + ((t-1)/(t-s))*(
  (1-s)*x1 + s*x2) by A2,A4,RLVECT_1:def 5
    .= (((-s+1)/(t-s))*((1-t)*x1) + ((-s+1)/(t-s))*(t*x2)) + (((t-1)/(t-s))*
  ((1-s)*x1) + ((t-1)/(t-s))*(s*x2)) by RLVECT_1:def 5
    .= ((((-s+1)/(t-s))*(1-t))*x1 + ((-s+1)/(t-s))*(t*x2)) + (((t-1)/(t-s))*
  ((1-s)*x1) + ((t-1)/(t-s))*(s*x2)) by RLVECT_1:def 7
    .= ((((-s+1)/(t-s))*(1-t))*x1 + (((-s+1)/(t-s))*t)*x2) + (((t-1)/(t-s))*
  ((1-s)*x1) + ((t-1)/(t-s))*(s*x2)) by RLVECT_1:def 7
    .= ((((-s+1)/(t-s))*(1-t))*x1 + (((-s+1)/(t-s))*t)*x2) + ((((t-1)/(t-s))
  *(1-s))*x1 + ((t-1)/(t-s))*(s*x2)) by RLVECT_1:def 7
    .= ((((-s+1)/(t-s))*(1-t))*x1 + (((-s+1)/(t-s))*t)*x2) + ((((t-1)/(t-s))
  *(1-s))*x1 + (((t-1)/(t-s))*s)*x2) by RLVECT_1:def 7
    .= (((-s+1)*(1-t))/(t-s))*x1 + ((((-s+1)*t)/(t-s))*x2 + ((((t-1)*(1-s))/
  (t-s))*x1 + (((t-1)*s)/(t-s))*x2)) by RLVECT_1:def 3
    .= (((-s+1)*(1-t))/(t-s))*x1 + ((((t-1)*(1-s))/(t-s))*x1 + ((((-s+1)*t)/
  (t-s))*x2 + (((t-1)*s)/(t-s))*x2)) by RLVECT_1:def 3
    .= ((((-s+1)*(1-t))/(t-s))*x1 + (((t-1)*(1-s))/(t-s))*x1) + ((((-s+1)*t)
  /(t-s))*x2 + (((t-1)*s)/(t-s))*x2) by RLVECT_1:def 3
    .= (((-s+1)*(1-t))/(t-s) + ((t-1)*(1-s))/(t-s))*x1 + ((((-s+1)*t)/(t-s))
  *x2 + (((t-1)*s)/(t-s))*x2) by RLVECT_1:def 6
    .= (((-s+1)*(1-t))/(t-s) + ((t-1)*(1-s))/(t-s))*x1 + (((-s+1)*t)/(t-s) +
  ((t-1)*s)/(t-s))*x2 by RLVECT_1:def 6
    .= 0.V + (((-s+1)*t + (t-1)*s)/(t-s))*x2 by RLVECT_1:10
    .= (1*(t-s)/(t-s))*x2
    .= 1*x2 by A5,XCMPLX_1:89
    .= x2 by RLVECT_1:def 8;
  then
A6: x2 in Line(y1,y2);
  (1-(t/(t-s)))*y1 + (t/(t-s))*y2 = ((1*(t-s)-t)/(t-s))*y1 + (t/(t-s))*y2
  by A5,XCMPLX_1:127
    .= (((-s)/(t-s))*((1-t)*x1) + ((-s)/(t-s))*(t*x2)) + (t/(t-s))*((1-s)*x1
  + s*x2) by A2,A4,RLVECT_1:def 5
    .= (((-s)/(t-s))*((1-t)*x1) + ((-s)/(t-s))*(t*x2)) + ((t/(t-s))*((1-s)*
  x1) + (t/(t-s))*(s*x2)) by RLVECT_1:def 5
    .= ((((-s)/(t-s))*(1-t))*x1 + ((-s)/(t-s))*(t*x2)) + ((t/(t-s))*((1-s)*
  x1) + (t/(t-s))*(s*x2)) by RLVECT_1:def 7
    .= ((((-s)/(t-s))*(1-t))*x1 + (((-s)/(t-s))*t)*x2) + ((t/(t-s))*((1-s)*
  x1) + (t/(t-s))*(s*x2)) by RLVECT_1:def 7
    .= ((((-s)/(t-s))*(1-t))*x1 + (((-s)/(t-s))*t)*x2) + (((t/(t-s))*(1-s))*
  x1 + (t/(t-s))*(s*x2)) by RLVECT_1:def 7
     .= ((((-s)/(t-s))*(1-t))*x1 + (((-s)/(t-s))*t)*x2) + (((t/(t-s))*(1-s))*
   x1 + ((t/(t-s))*s)*x2) by RLVECT_1:def 7
    .= (((-s)*(1-t))/(t-s))*x1 + ((((-s)*t)/(t-s))*x2 + (((t*(1-s))/(t-s))*
  x1 + ((t*s)/(t-s))*x2)) by RLVECT_1:def 3
    .= (((-s)*(1-t))/(t-s))*x1 + (((t*(1-s))/(t-s))*x1 + ((((-s)*t)/(t-s))*
  x2 + ((t*s)/(t-s))*x2)) by RLVECT_1:def 3
    .= ((((-s)*(1-t))/(t-s))*x1 + ((t*(1-s))/(t-s))*x1) + ((((-s)*t)/(t-s))*
  x2 + ((t*s)/(t-s))*x2) by RLVECT_1:def 3
    .= (((-s)*(1-t))/(t-s) + (t*(1-s))/(t-s))*x1 + ((((-s)*t)/(t-s))*x2 + ((
  t*s)/(t-s))*x2) by RLVECT_1:def 6
    .= (((-s)*(1-t)+t*(1-s))/(t-s))*x1 + (((-s)*t)/(t-s) + (t*s)/(t-s))*x2
                  by RLVECT_1:def 6
    .= (((-s)*(1-t)+t*(1-s))/(t-s))*x1 + 0.V by RLVECT_1:10
    .= ((1*(t-s))/(t-s))*x1
    .= 1*x1 by A5,XCMPLX_1:89
    .= x1 by RLVECT_1:def 8;
  then x1 in Line(y1,y2);
  hence thesis by A6,Th74;
end;
