
theorem Th10:
  for V, V9, C, C9 being set, A being Element of Fin PFuncs (V, C)
, B being Element of Fin PFuncs (V9, C9) st V c= V9 & C c= C9 & A = B holds mi
  A = mi B
proof
  let V, V9, C, C9 be set, A be Element of Fin PFuncs (V, C), B be Element of
  Fin PFuncs (V9, C9);
  assume that
A1: V c= V9 & C c= C9 and
A2: A = B;
  hereby
    let x be object;
A3: PFuncs (V,C) c= PFuncs (V9,C9) by A1,PARTFUN1:50;
    assume
A4: x in mi A;
    then x in { t where t is Element of PFuncs (V, C) : t is finite & for s
    being Element of PFuncs (V, C) holds ( s in A & s c= t iff s = t ) } by
SUBSTLAT:def 2;
    then consider f being Element of PFuncs (V, C) such that
A5: f = x and
A6: f is finite and
    for s being Element of PFuncs (V, C) holds ( s in A & s c= f iff s = f );
    reconsider f as Element of PFuncs (V9, C9) by A3;
    for s being Element of PFuncs (V9, C9) holds s in B & s c= f iff s = f
    by A2,A4,A5,SUBSTLAT:6;
    then x in { t where t is Element of PFuncs (V9, C9) : t is finite & for s
being Element of PFuncs (V9, C9) holds ( s in B & s c= t iff s = t ) } by A5,A6
    ;
    hence x in mi B by SUBSTLAT:def 2;
  end;
  let x be object;
  assume
A7: x in mi B;
  then x in { t where t is Element of PFuncs (V9, C9) : t is finite & for s
  being Element of PFuncs (V9, C9) holds ( s in B & s c= t iff s = t ) } by
SUBSTLAT:def 2;
  then ex f being Element of PFuncs (V9, C9) st f = x & f is finite & for s
  being Element of PFuncs (V9, C9) holds ( s in B & s c= f iff s = f );
  then reconsider x9 = x as finite set;
  mi B c= B & for b being finite set st b in A & b c= x9 holds b = x9 by A2,A7,
SUBSTLAT:6;
  hence thesis by A2,A7,SUBSTLAT:7;
end;
