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 Th10: :: AFF_1:27
  y1 in Line(x1,x2) & y2 in Line(x1,x2) implies Line(y1,y2) c= Line(x1,x2)
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;
    let z be object;
    assume z in Line(y1,y2);
    then consider u such that
A3: z = (1-u)*y1 + u*y2;
    z = ((1-u)*((1-t)*x1)+(1-u)*(t*x2))+u*((1-s)*x1+s*x2) by A1,A2,A3,
RVSUM_1:51
      .= ((1-u)*((1-t)*x1)+(1-u)*(t*x2))+(u*((1-s)*x1)+u*(s*x2)) by RVSUM_1:51
      .= (((1-u)*(1-t))*x1+(1-u)*(t*x2))+(u*((1-s)*x1)+u*(s*x2)) by RVSUM_1:49
      .= (((1-u)*(1-t))*x1+((1-u)*t)*x2)+(u*((1-s)*x1)+u*(s*x2)) by RVSUM_1:49
      .= (((1-u)*(1-t))*x1+((1-u)*t)*x2)+((u*(1-s))*x1+u*(s*x2)) by RVSUM_1:49
      .= (((1-u)*(1-t))*x1+((1-u)*t)*x2)+((u*(1-s))*x1+(u*s)*x2) by RVSUM_1:49
      .= ((1-u)*(1-t))*x1+(((1-u)*t)*x2+((u*(1-s))*x1+(u*s)*x2)) by FINSEQOP:28
      .= ((1-u)*(1-t))*x1+((u*(1-s))*x1+(((1-u)*t)*x2+(u*s)*x2)) by Lm2
      .= (((1-u)*(1-t))*x1+(u*(1-s))*x1)+(((1-u)*t)*x2+(u*s)*x2) by FINSEQOP:28
      .= ((1-u)*(1-t)+u*(1-s))*x1+(((1-u)*t)*x2+(u*s)*x2) by RVSUM_1:50
      .= (1-(1*t-u*t+u*s))*x1+(1*t-u*t+u*s)*x2 by RVSUM_1:50;
    hence z in Line(x1,x2);
end;
