reserve r, r1, r2, x, y, z,
        x1, x2, x3, y1, y2, y3 for Real;
reserve R, R1, R2, R3 for Element of 3-tuples_on REAL;
reserve p, q, p1, p2, p3, q1, q2 for Element of REAL 3;
reserve f1, f2, f3, g1, g2, g3, h1, h2, h3 for PartFunc of REAL,REAL;
reserve t, t0, t1, t2 for Real;

theorem
  (p = VFunc(f1,f2,f3).t1 & q = VFunc(g1,g2,g3).t2 & p = q) implies
  (f1.t1 = g1.t2 & f2.t1 = g2.t2 & f3.t1 = g3.t2)
proof
    assume
A1: p = VFunc(f1,f2,f3).t1 & q = VFunc(g1,g2,g3).t2 & p = q;
then A2: p.1 = f1.t1 & q.1 = g1.t2 by Th41;
A3: p.2 = f2.t1 & q.2 = g2.t2 by A1,Th41;
    p.3 = f3.t1 & q.3 = g3.t2 by A1,Th41;
    hence thesis by A1,A2,A3;
end;
