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 Th83:
  x1 in plane(y1,y2,y3) & x2 in plane(y1,y2,y3) & x3 in plane(y1,
  y2,y3) implies plane(x1,x2,x3) c= plane(y1,y2,y3)
proof
  assume that
A1: x1 in plane(y1,y2,y3) and
A2: x2 in plane(y1,y2,y3) and
A3: x3 in plane(y1,y2,y3);
  ex x29 be Element of REAL n st x2 = x29 &
  ex a21,a22,a23 being Real st
  a21 + a22 + a23 = 1 & x29 = a21 * y1 + a22 * y2 + a23 * y3 by A2;
  then consider a21,a22,a23 being Real such that
A4: a21 + a22 + a23 = 1 and
A5: x2 = a21*y1 + a22*y2 + a23*y3;
  ex x19 be Element of REAL n st x1 = x19 &
   ex a11,a12,a13 being Real st
  a11 + a12 + a13 = 1 & x19 = a11 * y1 + a12 * y2 + a13 * y3 by A1;
  then consider a11,a12,a13 being Real such that
A6: a11 + a12 + a13 = 1 and
A7: x1 = a11*y1 + a12*y2 + a13*y3;
  let x be object;
  assume x in plane(x1,x2,x3);
  then
  ex x9 be Element of REAL n st x = x9 &
  ex b1,b2,b3 being Real st b1 + b2
  + b3 = 1 & x9 = b1 * x1 + b2 * x2 + b3 * x3;
  then consider b1,b2,b3 being Real such that
A8: b1 + b2 + b3 = 1 and
A9: x = b1*x1 + b2*x2 + b3*x3;
  ex x39 be Element of REAL n st x3 = x39 &
  ex a31,a32,a33 being Real st
  a31 + a32 + a33 = 1 & x39 = a31 * y1 + a32 * y2 + a33 * y3 by A3;
  then consider a31,a32,a33 being Real such that
A10: a31 + a32 + a33 = 1 and
A11: x3 = a31*y1 + a32*y2 + a33*y3;
A12: (b1*a11 + b2*a21 + b3*a31) + (b1*a12 + b2*a22 + b3*a32) + (b1*a13 + b2*
  a23 + b3*a33) = b1*(a11 + a12 + a13) + b2*(a21 + a22 + a23)+ b3*(a31 + a32 +
  a33)
    .= 1 by A6,A4,A10,A8;
  x = ((b1*a11) * y1 + (b1*a12) * y2 + (b1*a13) * y3) + b2*(a21 * y1 +
  a22 * y2 + a23 * y3) + b3*(a31 * y1 + a32 * y2 + a33 * y3) by A7,A5,A11,A9
,Th22
    .= ((b1*a11) * y1 + (b1*a12) * y2 + (b1*a13) * y3) + ((b2*a21) * y1 + (
  b2*a22) * y2 + (b2*a23) * y3) + b3*(a31 * y1 + a32 * y2 + a33 * y3) by Th22
    .= ((b1*a11) * y1 + (b1*a12) * y2 + (b1*a13) * y3) + ((b2*a21) * y1 + (
b2*a22) * y2 + (b2*a23) * y3) + ((b3*a31) * y1 + (b3*a32) * y2 + (b3*a33) * y3)
  by Th22
    .= ((b1*a11+b2*a21) * y1 + (b1*a12+b2*a22) * y2 + (b1*a13+b2*a23) * y3)
  + ((b3*a31) * y1 + (b3*a32) * y2 + (b3*a33) * y3) by Th24
    .= (b1*a11 + b2*a21 + b3*a31)*y1 + (b1*a12 + b2*a22 + b3*a32)*y2 + (b1*
  a13 + b2*a23 + b3*a33)*y3 by Th24;
  hence thesis by A12;
end;
