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 Th41:
  p = VFunc(f1,f2,f3).t iff p.1 = f1.t & p.2 = f2.t & p.3 = f3.t
proof
  thus p = VFunc(f1,f2,f3).t implies p.1 = f1.t & p.2 = f2.t & p.3 = f3.t
  proof
    assume p = VFunc(f1,f2,f3).t;
    then p = |[ f1.t,f2.t,f3.t ]| by Def5;
    hence thesis;
  end;
  assume p.1 = f1.t & p.2 = f2.t & p.3 = f3.t;
  then p = |[ f1.t,f2.t,f3.t ]| by Th1;
  hence p = VFunc(f1,f2,f3).t by Def5;
end;
