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 Th58:
  dom f = [:X,Y:] & dom g = [:X9,Y9:] implies
  dom|:f,g:| = [:[:X,X9:],[:Y,Y9:]:]
proof
  assume
A1: dom f = [:X,Y:] & dom g = [:X9,Y9:];
  hence dom|:f,g:| c= [:[:X,X9:],[:Y,Y9:]:] by Th57;
  let z be object;
  assume z in [:[:X,X9:],[:Y,Y9:]:];
  then consider xx,yy being object such that
A2: xx in [:X,X9:] and
A3: yy in [:Y,Y9:] and
A4: z = [xx,yy] by ZFMISC_1:def 2;
  consider y,y9 being object such that
A5: y in Y & y9 in Y9 and
A6: yy = [y,y9] by A3,ZFMISC_1:def 2;
  consider x,x9 being object such that
A7: x in X & x9 in X9 and
A8: xx = [x,x9] by A2,ZFMISC_1:def 2;
  [x,y] in dom f & [x9,y9] in dom g by A1,A7,A5,ZFMISC_1:87;
  hence thesis by A4,A8,A6,Def3;
end;
