reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;

theorem Th29:
  [:{x},{y,z}:] = {[x,y],[x,z]} & [:{x,y},{z}:] = {[x,z],[y,z]}
proof
  now
    let v;
A1: z in{y,z} by TARSKI:def 2;
    thus v in [:{x},{y,z}:] implies v in {[x,y],[x,z]}
    proof
      assume v in [:{x},{y,z}:];
      then consider x1,y1 such that
A2:   x1 in {x} and
A3:   y1 in {y,z} and
A4:   v=[x1,y1] by Def2;
A5:   y1=y or y1=z by A3,TARSKI:def 2;
      x1=x by A2,TARSKI:def 1;
      hence thesis by A4,A5,TARSKI:def 2;
    end;
    assume v in {[x,y],[x,z]};
    then
A6: v=[x,y] or v=[x,z] by TARSKI:def 2;
    x in{x} & y in{y,z} by TARSKI:def 1,def 2;
    hence v in [:{x},{y,z}:] by A6,A1,Lm16;
  end;
  hence [:{x},{y,z}:] = {[x,y],[x,z]} by TARSKI:2;
  now
    let v;
A7: z in{z} by TARSKI:def 1;
    thus v in [:{x,y},{z}:] implies v in {[x,z],[y,z]}
    proof
      assume v in [:{x,y},{z}:];
      then consider x1,y1 such that
A8:   x1 in {x,y} and
A9:   y1 in {z} and
A10:  v=[x1,y1] by Def2;
A11:  x1=x or x1=y by A8,TARSKI:def 2;
      y1=z by A9,TARSKI:def 1;
      hence thesis by A10,A11,TARSKI:def 2;
    end;
    assume v in {[x,z],[y,z]};
    then
A12: v=[x,z] or v=[y,z] by TARSKI:def 2;
    x in{x,y} & y in{x,y} by TARSKI:def 2;
    hence v in [:{x,y},{z}:] by A12,A7,Lm16;
  end;
  hence thesis by TARSKI:2;
end;
