reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem
  for D,C be non empty set, F be finite PartFunc of D,REAL, G be finite
PartFunc of C,REAL st max+ F, max+ G are_fiberwise_equipotent & max- F, max- G
  are_fiberwise_equipotent holds F, G are_fiberwise_equipotent
proof
  let D,C be non empty set, F be finite PartFunc of D,REAL, G be finite
  PartFunc of C,REAL;
  assume that
A1: max+ F, max+ G are_fiberwise_equipotent and
A2: max- F, max- G are_fiberwise_equipotent;
  set lh = left_closed_halfline(0), rh = right_closed_halfline(0), fp0 = (max+
  F)"{0}, fm0 = (max- F)"{0}, gp0 = (max+ G)"{0}, gm0 = (max- G)"{0};
A3: lh /\ rh = [.0,0 .] by XXREAL_1:272
    .= {0} by XXREAL_1:17;
  F"(rng F) c= F"REAL by RELAT_1:143;
  then
A4: F"REAL c= dom F & dom F c= F"REAL by RELAT_1:132,134;
A5: F"lh = fp0 & F"rh =fm0 by Th36,Th39;
  G"(rng G) c= G"REAL by RELAT_1:143;
  then
A6: G"REAL c= dom G & dom G c= G"REAL by RELAT_1:132,134;
A7: G"lh = gp0 & G"rh =gm0 by Th36,Th39;
  reconsider fp0,fm0,gp0,gm0 as finite set;
A8: lh \/ rh = REAL \ ].0,0 .[ by XXREAL_1:398
    .= REAL \ {} by XXREAL_1:28
    .= REAL;
  then fp0 \/ fm0 = F"(REAL) by A5,RELAT_1:140;
  then
A9: fp0 \/ fm0 = dom F by A4;
  gp0 \/ gm0 = G"(lh \/ rh) by A7,RELAT_1:140;
  then
A10: gp0 \/ gm0 = dom G by A8,A6;
  card(fp0 \/ fm0) = card fp0 + card fm0 - card(fp0 /\ fm0) by CARD_2:45;
  then
A11: card(fp0 /\ fm0) = card fp0 + card fm0 - card(fp0 \/ fm0);
  card(gp0 \/ gm0) = card gp0 + card gm0 - card(gp0 /\ gm0) by CARD_2:45;
  then
A12: card(gp0 /\ gm0) = card gp0 + card gm0 - card(gp0 \/ gm0);
A13: dom F = dom(max+ F) & dom G = dom(max+ G) by Def10;
A14: now
    let r be Element of REAL;
A15: card fp0 = card gp0 & card fm0 = card gm0 by A1,A2,CLASSES1:78;
    now
      per cases;
      case
        0<r;
        then Coim(F,r) = Coim(max+ F,r) & Coim(G,r) = Coim(max+ G,r) by Th35;
        hence card Coim(F,r) = card Coim(G,r) by A1,CLASSES1:def 10;
      end;
      case
        0=r;
        then F"{r} = F"lh /\ F"rh & G"{r} = G"lh /\ G"rh by A3,FUNCT_1:68;
        hence card(F"{r}) = card(G"{r}) by A1,A13,A5,A7,A11,A12,A9,A10,A15,
CLASSES1:81;
      end;
      case
A16:    r<0;
A17:    --r=r;
        0<-r by A16,XREAL_1:58;
        then Coim(F,r) = Coim(max- F,-r) & Coim(G,r) = Coim(max- G,-r) by A17
,Th38;
        hence card Coim(F,r) = card Coim(G,r) by A2,CLASSES1:def 10;
      end;
    end;
    hence card Coim(F,r) = card Coim(G,r);
  end;
  rng F c= REAL & rng G c= REAL;
  hence thesis by A14,CLASSES1:79;
end;
