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 Th57:
  dom f c= [:X,Y:] & dom g c= [:X9,Y9:] implies dom|:f,g:| c= [:[:
  X,X9:],[:Y,Y9:]:]
proof
  assume
A1: dom f c= [:X,Y:] & dom g c= [:X9,Y9:];
  let xy be object;
  assume xy in dom|:f,g:|;
  then consider x,y,x9,y9 being object such that
A2: xy = [[x,x9],[y,y9]] and
A3: [x,y] in dom f & [x9,y9] in dom g by Def3;
  y in Y & y9 in Y9 by A1,A3,ZFMISC_1:87;
  then
A4: [y,y9] in [:Y,Y9:] by ZFMISC_1:87;
  x in X & x9 in X9 by A1,A3,ZFMISC_1:87;
  then [x,x9] in [:X,X9:] by ZFMISC_1:87;
  hence thesis by A2,A4,ZFMISC_1:87;
end;
