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 PartFunc of D,REAL, G be PartFunc of C,
  REAL st F,G are_fiberwise_equipotent holds max-(F), max-(G)
  are_fiberwise_equipotent
proof
  set li =right_closed_halfline(0);
  let D,C be non empty set, F be PartFunc of D,REAL, G be PartFunc of C,REAL;
  assume
A1: F,G are_fiberwise_equipotent;
A2: now
    let r be Element of REAL;
    now
      per cases;
      case
        0<r;
        then Coim(F,-r) = (max- F)"{r} & Coim(G,-r) = (max- G)"{r} by Th38;
        hence card Coim(max- F,r) = card Coim(max- G,r) by A1,CLASSES1:def 10;
      end;
      case
A3:     r=0;
        F" li = (max- F)"{0} & G" li = (max- G)"{0} by Th39;
        hence card((max- F)"{r}) = card((max- G)"{r}) by A1,A3,CLASSES1:78;
      end;
      case
A4:     r<0;
        now
          assume r in rng(max- F);
          then
ex d be Element of D st d in dom(max- F) & (max- F).d = r by PARTFUN1:3;
          hence contradiction by A4,Th40;
        end;
        then
A5:     (max- F)"{r} = {} by Lm2;
        now
          assume r in rng(max- G);
          then
ex c be Element of C st c in dom(max- G) & (max- G).c = r by PARTFUN1:3;
          hence contradiction by A4,Th40;
        end;
        hence card(( max-F)"{r}) = card(( max- G)"{r}) by A5,Lm2;
      end;
    end;
    hence card Coim(max-F,r) = card Coim(max- G,r);
  end;
  rng(max- F) c= REAL & rng(max- G) c= REAL;
  hence thesis by A2,CLASSES1:79;
end;
