reserve x, y, i for object,
  L for up-complete Semilattice;

theorem Th12:
  for L being non empty RelStr for F, G being Function-yielding
Function st dom F = dom G & (for x st x in dom F holds //\(F.x, L) = //\(G.x, L
  )) holds /\\(F, L) = /\\(G, L)
proof
  let L be non empty RelStr;
  let F, G be Function-yielding Function such that
A1: dom F = dom G and
A2: for x st x in dom F holds //\(F.x, L) = //\(G.x, L);
A3: for x being object st x in dom F holds /\\(F, L).x = /\\(G, L).x
  proof
    let x be object;
    assume
A4: x in dom F;
    hence /\\(F, L).x = //\(F.x, L) by Def2
      .= //\(G.x, L) by A2,A4
      .= /\\(G, L).x by A1,A4,Def2;
  end;
  dom /\\(F, L) = dom F & dom /\\(G, L) = dom G by FUNCT_2:def 1;
  hence thesis by A1,A3,FUNCT_1:2;
end;
