reserve a,b,s,t,u,lambda for Real,
  n for Nat;
reserve x,x1,x2,x3,y1,y2 for Element of REAL n;

theorem Th11: :: AFF_1:28
  y1 in Line(x1,x2) & y2 in Line(x1,x2) & y1<>y2 implies Line(x1,
  x2) c= Line(y1,y2)
proof
  assume y1 in Line(x1,x2);
  then consider t such that
A1: y1 = (1-t)*x1 + t*x2;
  assume y2 in Line(x1,x2);
  then consider s such that
A2: y2 = (1-s)*x1 + s*x2;
  assume y1<>y2;
  then
A3: t-s<>0 by A1,A2;
  then
  (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 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 A1,A2,RVSUM_1:51
    .= (((-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 RVSUM_1:51
    .= ((((-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 RVSUM_1:49
    .= ((((-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 RVSUM_1:49
    .= ((((-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 RVSUM_1:49
    .= ((((-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 RVSUM_1:49
    .= ((((-s+1)*(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 XCMPLX_1:99
    .= ((((-s+1)*(1-t))*(1/(t-s)))*x1 + (((-s+1)/(t-s))*t)*x2) + ((((t-1)/(t
  -s))*(1-s))*x1 + (((t-1)/(t-s))*s)*x2)
    .= ((((-s+1)*(1-t))/(t-s))*x1 + (((-s+1)/(t-s))*t)*x2) + ((((t-1)/(t-s))
  *(1-s))*x1 + (((t-1)/(t-s))*s)*x2) by XCMPLX_1:99
    .= ((((-s+1)*(1-t))/(t-s))*x1 + (((-s+1)*(1/(t-s)))*t)*x2) + ((((t-1)/(t
  -s))*(1-s))*x1 + (((t-1)/(t-s))*s)*x2) by XCMPLX_1:99
    .= ((((-s+1)*(1-t))/(t-s))*x1 + (((-s+1)*t)*(1/(t-s)))*x2) + ((((t-1)/(t
  -s))*(1-s))*x1 + (((t-1)/(t-s))*s)*x2)
    .= ((((-s+1)*(1-t))/(t-s))*x1 + (((-s+1)*t)/(t-s))*x2) + ((((t-1)/(t-s))
  *(1-s))*x1 + (((t-1)/(t-s))*s)*x2) by XCMPLX_1:99
    .= ((((-s+1)*(1-t))/(t-s))*x1 + (((-s+1)*t)/(t-s))*x2) + ((((t-1)*(1/(t-
  s)))*(1-s))*x1 + (((t-1)/(t-s))*s)*x2) by XCMPLX_1:99
    .= ((((-s+1)*(1-t))/(t-s))*x1 + (((-s+1)*t)/(t-s))*x2) + ((((t-1)*(1-s))
  *(1/(t-s)))*x1 + (((t-1)/(t-s))*s)*x2)
    .= ((((-s+1)*(1-t))/(t-s))*x1 + (((-s+1)*t)/(t-s))*x2) + ((((t-1)*(1-s))
  /(t-s))*x1 + (((t-1)/(t-s))*s)*x2) by XCMPLX_1:99
    .= ((((-s+1)*(1-t))/(t-s))*x1 + (((-s+1)*t)/(t-s))*x2) + ((((t-1)*(1-s))
  /(t-s))*x1 + (((t-1)*(1/(t-s)))*s)*x2) by XCMPLX_1:99
    .= ((((-s+1)*(1-t))/(t-s))*x1 + (((-s+1)*t)/(t-s))*x2) + ((((t-1)*(1-s))
  /(t-s))*x1 + (((t-1)*s)*(1/(t-s)))*x2)
    .= ((((-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 XCMPLX_1:99
    .= (((-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 FINSEQOP:28
    .= (((-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 Lm2
    .= ((((-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 FINSEQOP:28
    .= (((-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 RVSUM_1:50
    .= (((-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 RVSUM_1:50
    .= (((-s+1)*(1-t)+(t-1)*(1-s))/(t-s))*x1 + (((-s+1)*t)/(t-s) + ((t-1)*s)
  /(t-s))*x2 by XCMPLX_1:62
    .= (((1-t)*(1-s)+-(1-t)*(1-s))/(t-s))*x1 + (((-s+1)*t + (t-1)*s)/(t-s))*
  x2 by XCMPLX_1:62
    .= 0*n + (((-s+1)*t + (t-1)*s)/(t-s))*x2 by Th3
    .= (1*(t-s)/(t-s))*x2 by Th1
    .= 1*x2 by A3,XCMPLX_1:89
    .= x2 by Th3;
  then
A4: 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 A3,XCMPLX_1:127
    .= (((-s)/(t-s))*((1-t)*x1) + ((-s)/(t-s))*(t*x2)) + (t/(t-s))*((1-s)*x1
  + s*x2) by A1,A2,RVSUM_1:51
    .= (((-s)/(t-s))*((1-t)*x1) + ((-s)/(t-s))*(t*x2)) + ((t/(t-s))*((1-s)*
  x1) + (t/(t-s))*(s*x2)) by RVSUM_1:51
    .= ((((-s)/(t-s))*(1-t))*x1 + ((-s)/(t-s))*(t*x2)) + ((t/(t-s))*((1-s)*
  x1) + (t/(t-s))*(s*x2)) by RVSUM_1:49
    .= ((((-s)/(t-s))*(1-t))*x1 + (((-s)/(t-s))*t)*x2) + ((t/(t-s))*((1-s)*
  x1) + (t/(t-s))*(s*x2)) by RVSUM_1:49
    .= ((((-s)/(t-s))*(1-t))*x1 + (((-s)/(t-s))*t)*x2) + (((t/(t-s))*(1-s))*
  x1 + (t/(t-s))*(s*x2)) by RVSUM_1:49
    .= ((((-s)/(t-s))*(1-t))*x1 + (((-s)/(t-s))*t)*x2) + (((t/(t-s))*(1-s))*
  x1 + ((t/(t-s))*s)*x2) by RVSUM_1:49
    .= ((((-s)*(1/(t-s)))*(1-t))*x1 + (((-s)/(t-s))*t)*x2) + (((t/(t-s))*(1-
  s))*x1 + ((t/(t-s))*s)*x2) by XCMPLX_1:99
    .= ((((-s)*(1-t))*(1/(t-s)))*x1 + (((-s)/(t-s))*t)*x2) + (((t/(t-s))*(1-
  s))*x1 + ((t/(t-s))*s)*x2)
    .= ((((-s)*(1-t))/(t-s))*x1 + (((-s)/(t-s))*t)*x2) + (((t/(t-s))*(1-s))*
  x1 + ((t/(t-s))*s)*x2) by XCMPLX_1:99
    .= ((((-s)*(1-t))/(t-s))*x1 + (((-s)*(1/(t-s)))*t)*x2) + (((t/(t-s))*(1-
  s))*x1 + ((t/(t-s))*s)*x2) by XCMPLX_1:99
    .= ((((-s)*(1-t))/(t-s))*x1 + (((-s)*t)*(1/(t-s)))*x2) + (((t/(t-s))*(1-
  s))*x1 + ((t/(t-s))*s)*x2)
    .= ((((-s)*(1-t))/(t-s))*x1 + (((-s)*t)/(t-s))*x2) + (((t/(t-s))*(1-s))*
  x1 + ((t/(t-s))*s)*x2) by XCMPLX_1:99
    .= ((((-s)*(1-t))/(t-s))*x1 + (((-s)*t)/(t-s))*x2) + (((t*(1/(t-s)))*(1-
  s))*x1 + ((t/(t-s))*s)*x2) by XCMPLX_1:99
    .= ((((-s)*(1-t))/(t-s))*x1 + (((-s)*t)/(t-s))*x2) + (((t*(1-s))*(1/(t-s
  )))*x1 + ((t/(t-s))*s)*x2)
    .= ((((-s)*(1-t))/(t-s))*x1 + (((-s)*t)/(t-s))*x2) + (((t*(1-s))/(t-s))*
  x1 + ((t/(t-s))*s)*x2) by XCMPLX_1:99
    .= ((((-s)*(1-t))/(t-s))*x1 + (((-s)*t)/(t-s))*x2) + (((t*(1-s))/(t-s))*
  x1 + ((t*(1/(t-s)))*s)*x2) by XCMPLX_1:99
    .= ((((-s)*(1-t))/(t-s))*x1 + (((-s)*t)/(t-s))*x2) + (((t*(1-s))/(t-s))*
  x1 + ((t*s)*(1/(t-s)))*x2)
    .= ((((-s)*(1-t))/(t-s))*x1 + (((-s)*t)/(t-s))*x2) + (((t*(1-s))/(t-s))*
  x1 + ((t*s)/(t-s))*x2) by XCMPLX_1:99
    .= (((-s)*(1-t))/(t-s))*x1 + ((((-s)*t)/(t-s))*x2 + (((t*(1-s))/(t-s))*
  x1 + ((t*s)/(t-s))*x2)) by FINSEQOP:28
    .= (((-s)*(1-t))/(t-s))*x1 + (((t*(1-s))/(t-s))*x1 + ((((-s)*t)/(t-s))*
  x2 + ((t*s)/(t-s))*x2)) by Lm2
    .= ((((-s)*(1-t))/(t-s))*x1 + ((t*(1-s))/(t-s))*x1) + ((((-s)*t)/(t-s))*
  x2 + ((t*s)/(t-s))*x2) by FINSEQOP:28
    .= (((-s)*(1-t))/(t-s) + (t*(1-s))/(t-s))*x1 + ((((-s)*t)/(t-s))*x2 + ((
  t*s)/(t-s))*x2) by RVSUM_1:50
    .= (((-s)*(1-t))/(t-s) + (t*(1-s))/(t-s))*x1 + (((-s)*t)/(t-s) + (t*s)/(
  t-s))*x2 by RVSUM_1:50
    .= (((-s)*(1-t)+t*(1-s))/(t-s))*x1 + (((-s)*t)/(t-s) + (t*s)/(t-s))*x2
  by XCMPLX_1:62
    .= (((-s)*(1-t)+t*(1-s))/(t-s))*x1 + (((-s)*t + t*s)/(t-s))*x2 by
XCMPLX_1:62
    .= (((-s)*(1-t)+t*(1-s))/(t-s))*x1 + 0*n by Th3
    .= ((1*(t-s))/(t-s))*x1 by Th1
    .= 1*x1 by A3,XCMPLX_1:89
    .= x1 by Th3;
  then x1 in Line(y1,y2);
  hence thesis by A4,Th10;
end;
