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

theorem Th11:
  for D,C be non empty set, F be PartFunc of D,REAL, G be PartFunc
of C,REAL, r st r <> 0 holds F,G are_fiberwise_equipotent iff r(#)F, r
  (#)G are_fiberwise_equipotent
proof
  let D,C be non empty set, F be PartFunc of D,REAL, G be PartFunc of C,REAL,
  r;
  assume
A1: r <> 0;
   reconsider rr=r as Element of REAL by XREAL_0:def 1;
A2: rng(rr(#)F) c= REAL & rng(rr(#)G) c= REAL;
  thus F,G are_fiberwise_equipotent implies r(#)F, r(#) G
  are_fiberwise_equipotent
  proof
    assume
A3: F,G are_fiberwise_equipotent;
    now
      let x be Element of REAL;
      Coim(F,x/r) = Coim(r(#)F,x) & Coim(G,x/r) = Coim(r(#)G,x) by A1,Th6;
      hence card Coim(r(#)F,x) = card Coim(r(#)G,x) by A3,CLASSES1:def 10;
    end;
    hence thesis by A2,CLASSES1:79;
  end;
  assume
A4: r(#)F, r(#)G are_fiberwise_equipotent;
A5: now
    let x be Element of REAL;
A6: G"{r*x/r} = Coim(r(#)G,r*x) by A1,Th6;
    r*x/r = x & F"{r*x/r} = Coim(r(#)F,r*x) by A1,Th6,XCMPLX_1:89;
    hence card Coim(F,x) = card Coim(G,x) by A4,A6,CLASSES1:def 10;
  end;
  rng F c= REAL & rng G c= REAL;
  hence thesis by A5,CLASSES1:79;
end;
