reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;

theorem Th56:
  rng |:f,g:| c= [:rng f,rng g:]
proof
  let z be object;
  assume z in rng |:f,g:|;
  then consider p being object such that
A1: p in dom |:f,g:| and
A2: z = |:f,g:|.p by FUNCT_1:def 3;
  consider x,y,x9,y9 being object such that
A3: p = [[x,x9],[y,y9]] and
A4: [x,y] in dom f & [x9,y9] in dom g by A1,Def3;
A5: f.[x,y] in rng f & g.[x9,y9] in rng g by A4,FUNCT_1:def 3;
  z = |:f,g:|.([x,x9],[y,y9]) by A2,A3
    .= [f.(x,y),g.(x9,y9)] by A4,Def3;
  hence thesis by A5,ZFMISC_1:87;
end;
