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 Th85:
  y1 in plane(x1,x2,x3) & y2 in plane(x1,x2,x3) implies Line(y1,y2
  ) c= plane(x1,x2,x3)
proof
  assume that
A1: y1 in plane(x1,x2,x3) and
A2: y2 in plane(x1,x2,x3);
  consider y29 be Element of REAL n such that
A3: y2 = y29 and
A4: ex a21,a22,a23 being Real st a21 + a22 + a23 = 1 & y29 = a21 * x1 +
  a22 * x2 + a23 * x3 by A2;
  consider y19 be Element of REAL n such that
A5: y1 = y19 and
A6: ex a11,a12,a13 being Real st a11 + a12 + a13 = 1 & y19 = a11 * x1 +
  a12 * x2 + a13 * x3 by A1;
  Line(y1,y2) c= plane(x1,x2,x3)
  proof
    let x be object;
    assume x in Line(y1,y2);
    then consider t such that
A7: x = (1-t)*y1 + t*y2;
    consider a21,a22,a23 being Real such that
A8: a21 + a22 + a23 = 1 and
A9: y29 = a21 * x1 + a22 * x2 + a23 * x3 by A4;
    consider a11,a12,a13 being Real such that
A10: a11 + a12 + a13 = 1 and
A11: y19 = a11 * x1 + a12 * x2 + a13 * x3 by A6;
A12: ((1-t)*a11+t*a21) + ((1-t)*a12+t*a22) + ((1-t)*a13+t*a23) = (1-t)*(
    a11 + a12 + a13) + t*(a21 + a22 + a23)
      .= (1-t) + t by A10,A8
      .= 1;
    x = (((1-t)*a11)*x1 + ((1-t)*a12)*x2 + ((1-t)*a13)*x3) + t*(a21 * x1
    + a22 * x2 + a23 * x3) by A5,A3,A7,A11,A9,Th22
      .= (((1-t)*a11)*x1 + ((1-t)*a12)*x2 + ((1-t)*a13)*x3) + ((t*a21)*x1 +
    (t*a22)*x2 + (t*a23)*x3) by Th22
      .= ((1-t)*a11 + t*a21) * x1 + ((1-t)*a12 + t*a22) * x2 + ((1-t)*a13+t*
    a23) * x3 by Th24;
    hence thesis by A12;
  end;
  hence thesis;
end;
