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
  for A being Subset of REAL n,x,x1,x2,x3 st x in plane(x1,x2,x3) & ex
  c1,c2,c3 being Real st c1 + c2 + c3 = 0 & x = c1*x1 + c2*x2 + c3*x3 holds 0*n
  in plane(x1,x2,x3)
proof
  let A being Subset of REAL n,x,x1,x2,x3;
  assume that
A1: x in plane(x1,x2,x3) and
A2: ex c1,c2,c3 being Real st c1 + c2 + c3 = 0 & x = c1*x1 + c2*x2 + c3* x3;
  consider c1,c2,c3 being Real such that
A3: c1 + c2 + c3 = 0 and
A4: x = c1*x1 + c2*x2 + c3*x3 by A2;
  ex x9 be Element of REAL n st x = x9 &
  ex a1,a2,a3 being Real st a1 + a2
  + a3 = 1 & x9 = a1 * x1 + a2 * x2 + a3 * x3 by A1;
  then consider a1,a2,a3 being Real such that
A5: a1 + a2 + a3 = 1 and
A6: x = a1 * x1 + a2 * x2 + a3 * x3;
A7: (a1+ -c1) + (a2 + -c2) + (a3 + -c3) = 1 by A5,A3;
  0*n = x - x by Th2
    .= (a1*x1 + a2*x2 + a3*x3) + ((-c1)*x1 + (-c2)*x2 + (-c3)*x3) by A6,A4,Th14
    .= (a1 +-c1)*x1 + (a2+-c2)*x2 + (a3+-c3)*x3 by Th24;
  hence thesis by A7;
end;
