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 abs(F), abs(G)
  are_fiberwise_equipotent
proof
  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 abs(F)"{r} = F"{-r,r} & abs(G)"{r} = G"{-r,r} by Th8;
        hence card(abs(F)"{r}) = card(abs(G)"{r}) by A1,CLASSES1:78;
      end;
      case
        0=r;
        then abs(F)"{r} = F"{r} & abs(G)"{r} = G"{r} by Th9;
        hence card(abs(F)"{r}) = card(abs(G)"{r}) by A1,CLASSES1:78;
      end;
      case
A3:     r<0;
        then abs(F)"{r} = {} by Th10;
        hence card(abs(F)"{r}) = card(abs(G)"{r}) by A3,Th10;
      end;
    end;
    hence card Coim(abs(F),r) = card Coim(abs(G),r);
  end;
  rng abs(F) c= REAL & rng abs(G) c= REAL;
  hence thesis by A2,CLASSES1:79;
end;
