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 Th88:
  x in plane(x1,x2,x3) iff ex a1,a2,a3 st a1+a2+a3=1 & x = a1*x1+ a2*x2+a3*x3
proof
  thus x in plane(x1,x2,x3)
   implies ex a1,a2,a3 st a1+a2+a3=1 & x = a1*x1+a2*x2+a3*x3
  proof
    assume x in plane(x1,x2,x3);
    then
    ex x9 be Element of REAL n st x = x9 &
      ex a19,a29,a39 being Real st a19
      +a29+a39=1 & x9=a19*x1+a29*x2+a39*x3;
    hence thesis;
  end;
  thus thesis;
end;
