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 , r be Real holds F, G are_fiberwise_equipotent iff F-r,G-r
  are_fiberwise_equipotent
proof
  let D,C be non empty set, F be PartFunc of D,REAL, G be PartFunc of C,REAL,
  r be Real;
A1: rng(F-r) c= REAL & rng(G-r) c= REAL;
  thus F, G are_fiberwise_equipotent implies F-r, G-r are_fiberwise_equipotent
  proof
    assume
A2: F, G are_fiberwise_equipotent;
    now
      let s be Element of REAL;
      thus card Coim(F-r,s) = card Coim(F,s+r) by Th50
        .= card Coim(G,s+r) by A2,CLASSES1:def 10
        .= card Coim(G-r,s) by Th50;
    end;
    hence thesis by A1,CLASSES1:79;
  end;
  assume
A3: F-r, G-r are_fiberwise_equipotent;
A4: now
    let s be Element of REAL;
A5: s = s-r+r;
    hence card Coim(F,s) = card Coim(F-r,s-r) by Th50
      .= card Coim(G- r,s-r) by A3,CLASSES1:def 10
      .= card Coim(G,s) by A5,Th50;
  end;
  rng F c= REAL & rng G c= REAL;
  hence thesis by A4,CLASSES1:79;
end;
