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 Th82:
  x1 in plane(x1,x2,x3) & x2 in plane(x1,x2,x3) & x3 in plane(x1, x2,x3)
proof
A1: 0 * x3 = 0*n by EUCLID_4:3;
  1 * x1 = x1 & 0 * x2 = 0*n by EUCLID_4:3;
  then
A2: 1 * x1 + 0 * x2 + 0 * x3 = x1 + 0*n by A1,EUCLID_4:1
    .= x1 by EUCLID_4:1;
A3: 0 * x3 = 0*n by EUCLID_4:3;
A4: 1 * x3 = x3 by EUCLID_4:3;
  0 * x1 = 0*n & 0 * x2 = 0*n by EUCLID_4:3;
  then
A5: 0 * x1 + 0 * x2 + 1 * x3 = 0*n + x3 by A4,EUCLID_4:1
    .= x3 by EUCLID_4:1;
  0 * x1 = 0*n & 1 * x2 = x2 by EUCLID_4:3;
  then
A6: 0 * x1 + 1 * x2 + 0 * x3 = x2 + 0*n by A3,EUCLID_4:1
    .= x2 by EUCLID_4:1;
  0+0+1=1;
  hence thesis by A2,A6,A5;
end;
