
theorem Th1:
  for I be set, J be non empty set, f be Function of I,J*, X be
ManySortedSet of J, p be Element of J*, x be set st x in I & p = f.x holds (X#
  * f).x = product (X * p)
proof
  let I be set, J be non empty set, f be Function of I,J*, X be ManySortedSet
  of J, p be Element of J*, x be set;
  assume
A1: x in I & p = f.x;
A2: dom f = I by FUNCT_2:def 1;
  then dom (X# * f) = dom f by PARTFUN1:def 2;
  hence (X# * f).x =(X# qua ManySortedSet of J*).p by A1,A2,FUNCT_1:12
    .= product (X * p) by FINSEQ_2:def 5;
end;
