
theorem Th60:
  for A,B be Element of Family_of_Intervals
   st A misses B & A \/ B is Interval
     holds pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
proof
    let A,B be Element of Family_of_Intervals;
    assume that
A1:  A misses B and
A2:  A \/ B is Interval;
    A in the set of all I where I is Interval by MEASUR10:def 1; then
A3: ex I be Interval st A = I;
    B in the set of all I where I is Interval by MEASUR10:def 1; then
A4: ex I be Interval st B = I;
    per cases;
    suppose A5: A = {}; then
     pre-Meas.A = 0 by Th58,MEASURE5:10;
     hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B by A5,XXREAL_3:4;
    end;
    suppose A6: B = {}; then
     pre-Meas.B = 0 by Th58,MEASURE5:10;
     hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B by A6,XXREAL_3:4;
    end;
    suppose A7: A <> {} & B <> {};
      per cases by A3,MEASURE5:1;
      suppose A is closed_interval; then
A8:    A = [.inf A,sup A.] by A7,MEASURE6:17;
       inf A <= sup A by A7,A8,XXREAL_1:29; then
A9:   A is left_end right_end by A8,XXREAL_2:33;
A10:   now assume B is closed_interval; then
        B = [.inf B,sup B.] by A7,MEASURE6:17;
        hence contradiction by A1,A2,A7,A8,Th14;
       end;
       per cases by A4,A10,MEASURE5:1;
       suppose B is right_open_interval; then
        B = [.inf B,sup B.[ by A7,MEASURE6:18; then
A11:    inf A = sup B & A \/ B = [.inf B,sup A.]
          by A1,A2,A7,A8,Th15; then
A12:    sup(A \/ B) = sup A & inf(A \/ B) = inf B
          by A7,XXREAL_1:29,MEASURE6:10,14;
        pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A13:    pre-Meas.(A \/ B) = sup A - inf B by A7,A12,MEASURE5:def 6;
        pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
        pre-Meas.A = sup A - inf A
      & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
        hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
          by A13,A9,A11,XXREAL_3:34;
       end;
       suppose B is left_open_interval; then
        B = ].inf B,sup B.] by A7,MEASURE6:19; then
A14:    sup A = inf B & A \/ B = [.inf A,sup B.]
          by A1,A2,A7,A8,Th16; then
A15:    sup(A \/ B) = sup B & inf(A \/ B) = inf A
          by A7,XXREAL_1:29,MEASURE6:10,14;
        pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A16:    pre-Meas.(A \/ B) = sup B - inf A by A7,A15,MEASURE5:def 6;
        pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
        pre-Meas.A = sup A - inf A
      & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
        hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
          by A16,A9,A14,XXREAL_3:34;
       end;
       suppose B is open_interval; then
A17:    B = ].inf B,sup B.[ by A7,MEASURE6:16;
        per cases by A1,A2,A7,A8,A17,Th17;
        suppose
A18:     inf A = sup B & A \/ B = ].inf B,sup A.]; then
         inf B <= sup A by A7,XXREAL_1:26; then
A19:     sup(A \/ B) = sup A & inf(A \/ B) = inf B by A18,A7,MEASURE6:9,13;
         pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A20:     pre-Meas.(A \/ B) = sup A - inf B by A7,A19,MEASURE5:def 6;
         pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
         pre-Meas.A = sup A - inf A
       & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
         hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
           by A20,A9,A18,XXREAL_3:34;
        end;
        suppose
A21:     inf B = sup A & A \/ B = [.inf A,sup B.[; then
         inf A <= sup B by A7,XXREAL_1:27; then
A22:     sup(A \/ B) = sup B & inf(A \/ B) = inf A by A21,A7,MEASURE6:11,15;
         pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A23:     pre-Meas.(A \/ B) = sup B - inf A by A7,A22,MEASURE5:def 6;
         pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
         pre-Meas.A = sup A - inf A
       & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
         hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
           by A23,A9,A21,XXREAL_3:34;
        end;
       end;
      end;
      suppose A is right_open_interval; then
A24:    A = [.inf A,sup A.[ by A7,MEASURE6:18;
A25:   A is left_end by A7,A24,XXREAL_1:27,XXREAL_2:34;
A26:   now assume B is left_open_interval; then
        B = ].inf B,sup B.] by A7,MEASURE6:19;
        hence contradiction by A1,A2,A7,A24,Th19;
       end;
       per cases by A4,A26,MEASURE5:1;
       suppose B is closed_interval; then
A27:    B = [.inf B,sup B.] by A7,MEASURE6:17; then
A28:    inf B = sup A & A \/ B = [.inf A,sup B.]
          by A1,A2,A7,A24,Th15;
        inf B <= sup B by A7,A27,XXREAL_1:29; then
A29:    B is left_end right_end by A27,XXREAL_2:33;
A30:    sup(A \/ B) = sup B & inf(A \/ B) = inf A
          by A28,A7,XXREAL_1:29,MEASURE6:10,14;
        pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A31:    pre-Meas.(A \/ B) = sup B - inf A by A7,A30,MEASURE5:def 6;
        pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
        pre-Meas.A = sup A - inf A
      & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
        hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
          by A31,A29,A28,XXREAL_3:34;
       end;
       suppose B is right_open_interval; then
A32:    B = [.inf B,sup B.[ by A7,MEASURE6:18;
        per cases by A1,A2,A7,A24,A32,Th18;
        suppose
A33:     inf A = sup B & A \/ B = [.inf B,sup A.[; then
         inf B <= sup A by A7,XXREAL_1:27; then
A34:     sup(A \/ B) = sup A & inf(A \/ B) = inf B by A33,A7,MEASURE6:11,15;
         pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A35:     pre-Meas.(A \/ B) = sup A - inf B by A7,A34,MEASURE5:def 6;
         pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
         pre-Meas.A = sup A - inf A
       & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
         hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
           by A35,A25,A33,XXREAL_3:34;
        end;
        suppose
A36:     inf B = sup A & A \/ B = [.inf A,sup B.[;
A37:     B is left_end by A7,A32,XXREAL_1:27,XXREAL_2:34;
         inf A <= sup B by A36,A7,XXREAL_1:27; then
A38:     sup(A \/ B) = sup B & inf(A \/ B) = inf A by A36,A7,MEASURE6:11,15;
         pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A39:     pre-Meas.(A \/ B) = sup B - inf A by A7,A38,MEASURE5:def 6;
         pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
         pre-Meas.A = sup A - inf A
       & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
         hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
           by A39,A37,A36,XXREAL_3:34;
        end;
       end;
       suppose B is open_interval; then
        B = ].inf B,sup B.[ by A7,MEASURE6:16; then
A40:    sup B = inf A & A \/ B = ].inf B,sup A.[
          by A1,A2,A7,A24,Th20; then
        inf B <= sup A by A7,XXREAL_1:28; then
A41:    sup(A \/ B) = sup A & inf(A \/ B) = inf B by A40,A7,MEASURE6:8,12;
        pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A42:    pre-Meas.(A \/ B) = sup A - inf B by A7,A41,MEASURE5:def 6;
        pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
        pre-Meas.A = sup A - inf A
      & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
        hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
          by A42,A25,A40,XXREAL_3:34;
       end;
      end;
      suppose A is left_open_interval; then
A43:    A = ].inf A,sup A.] by A7,MEASURE6:19;
A44:   A is right_end by A7,A43,XXREAL_1:26,XXREAL_2:35;
A45:   now assume B is right_open_interval; then
        B = [.inf B,sup B.[ by A7,MEASURE6:18;
        hence contradiction by A1,A2,A7,A43,Th19;
       end;
       per cases by A4,A45,MEASURE5:1;
       suppose B is closed_interval; then
A46:    B = [.inf B,sup B.] by A7,MEASURE6:17;
        inf B <= sup B by A7,A46,XXREAL_1:29; then
A47:    B is left_end right_end by A46,XXREAL_2:33;
A48:    inf A = sup B & A \/ B = [.inf B,sup A.]
          by A1,A2,A7,A43,A46,Th16; then
A49:    sup(A \/ B) = sup A & inf(A \/ B) = inf B
          by A7,XXREAL_1:29,MEASURE6:10,14;
        pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A50:    pre-Meas.(A \/ B) = sup A - inf B by A7,A49,MEASURE5:def 6;
        pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
        pre-Meas.A = sup A - inf A
      & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
        hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
          by A50,A47,A48,XXREAL_3:34;
       end;
       suppose B is left_open_interval; then
A51:    B = ].inf B,sup B.] by A7,MEASURE6:19;
A52:    B is right_end by A7,A51,XXREAL_1:26,XXREAL_2:35;
        per cases by A1,A2,A7,A43,A51,Th21;
        suppose
A53:     inf A = sup B & A \/ B = ].inf B,sup A.]; then
         inf B <= sup A by A7,XXREAL_1:26; then
A54:     sup(A \/ B) = sup A & inf(A \/ B) = inf B by A53,A7,MEASURE6:9,13;
         pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A55:     pre-Meas.(A \/ B) = sup A - inf B by A7,A54,MEASURE5:def 6;
         pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
         pre-Meas.A = sup A - inf A
       & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
         hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
           by A55,A52,A53,XXREAL_3:34;
        end;
        suppose
A56:     inf B = sup A & A \/ B = ].inf A,sup B.]; then
         inf A <= sup B by A7,XXREAL_1:26; then
A57:     sup(A \/ B) = sup B & inf(A \/ B) = inf A by A56,A7,MEASURE6:9,13;
         pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A58:     pre-Meas.(A \/ B) = sup B - inf A by A7,A57,MEASURE5:def 6;
         pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
         pre-Meas.A = sup A - inf A
       & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
         hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
           by A58,A44,A56,XXREAL_3:34;
        end;
       end;
       suppose B is open_interval; then
         B = ].inf B,sup B.[ by A7,MEASURE6:16; then
A60:    inf B = sup A & A \/ B = ].inf A,sup B.[
          by A1,A2,A7,A43,Th22; then
        inf A <= sup B by A7,XXREAL_1:28; then
A61:    sup(A \/ B) = sup B & inf(A \/ B) = inf A by A60,A7,MEASURE6:8,12;
        pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A62:    pre-Meas.(A \/ B) = sup B - inf A by A7,A61,MEASURE5:def 6;
        pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
        pre-Meas.A = sup A - inf A
      & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
        hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
          by A62,A44,A60,XXREAL_3:34;
       end;
      end;
      suppose A is open_interval; then
A63:    A = ].inf A,sup A.[ by A7,MEASURE6:16;
A64:   now assume B is open_interval; then
        B = ].inf B,sup B.[ by A7,MEASURE6:16;
        hence contradiction by A1,A2,A7,A63,Th23;
       end;
       per cases by A4,A64,MEASURE5:1;
       suppose B is closed_interval; then
A65:    B = [.inf B,sup B.] by A7,MEASURE6:17;
        inf B <= sup B by A7,A65,XXREAL_1:29; then
A66:    B is left_end right_end by A65,XXREAL_2:33;
        per cases by A1,A2,A7,A63,A65,Th17;
        suppose
A67:     inf A = sup B & A \/ B = [.inf B,sup A.[; then
         inf B <= sup A by A7,XXREAL_1:27; then
A68:     sup(A \/ B) = sup A & inf(A \/ B) = inf B by A67,A7,MEASURE6:11,15;
         pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A69:     pre-Meas.(A \/ B) = sup A - inf B by A7,A68,MEASURE5:def 6;
         pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
         pre-Meas.A = sup A - inf A
       & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
         hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
           by A69,A67,A66,XXREAL_3:34;
        end;
        suppose
A70:     inf B = sup A & A \/ B = ].inf A,sup B.]; then
         inf A <= sup B by A7,XXREAL_1:26; then
A71:     sup(A \/ B) = sup B & inf(A \/ B) = inf A by A70,A7,MEASURE6:9,13;
         pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A72:     pre-Meas.(A \/ B) = sup B - inf A by A7,A71,MEASURE5:def 6;
         pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
         pre-Meas.A = sup A - inf A
       & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
         hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
           by A72,A70,A66,XXREAL_3:34;
        end;
       end;
       suppose B is left_open_interval; then
A73:    B = ].inf B,sup B.] by A7,MEASURE6:19;
A74:    sup B = inf A & A \/ B = ].inf B,sup A.[
          by A1,A2,A7,A63,A73,Th22; then
        inf B <= sup A by A7,XXREAL_1:28; then
A75:    sup(A \/ B) = sup A & inf(A \/ B) = inf B by A74,A7,MEASURE6:8,12;
A76:    B is right_end by A7,A73,XXREAL_1:26,XXREAL_2:35;
        pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A77:    pre-Meas.(A \/ B) = sup A - inf B by A7,A75,MEASURE5:def 6;
        pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
        pre-Meas.A = sup A - inf A
      & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
        hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
          by A77,A74,A76,XXREAL_3:34;
       end;
       suppose B is right_open_interval; then
A78:    B = [.inf B,sup B.[ by A7,MEASURE6:18; then
A79:    sup A = inf B & A \/ B = ].inf A,sup B.[
          by A1,A2,A7,A63,Th20; then
        inf A <= sup B by A7,XXREAL_1:28; then
A80:    sup(A \/ B) = sup B & inf(A \/ B) = inf A by A79,A7,MEASURE6:8,12;
A81:    B is left_end by A7,A78,XXREAL_1:27,XXREAL_2:34;
        pre-Meas.(A \/ B) = diameter(A \/ B) by A2,Th59; then
A82:    pre-Meas.(A \/ B) = sup B - inf A by A7,A80,MEASURE5:def 6;
        pre-Meas.A = diameter A & pre-Meas.B = diameter B by Th58; then
        pre-Meas.A = sup A - inf A
      & pre-Meas.B = sup B - inf B by A7,MEASURE5:def 6;
        hence pre-Meas.(A \/ B) = pre-Meas.A + pre-Meas.B
          by A82,A79,A81,XXREAL_3:34;
       end;
      end;
    end;
end;
