reserve a, b, p, q for Real;

theorem Th5:
  1 < p & 1/p + 1/q = 1 & 0 <= a & 0 <= b implies a * b <= a
to_power p / p + b to_power q / q & (a * b = a to_power p / p + b to_power q /
  q iff a to_power p = b to_power q)
proof
  assume that
A1: 1 < p and
A2: 1/p + 1/q = 1 and
A3: 0 <= a and
A4: 0 <= b;
A5: 0 <= a to_power p / p
  proof
    now
      per cases by A3;
      case
        0 < a;
        then 0 < a to_power p by POWER:34;
        hence thesis by A1;
      end;
      case
        0 = a;
        then 0 = a to_power p by A1,POWER:def 2;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  reconsider pp=1/p as Real;
  1/p < 1 by A1,XREAL_1:189;
  then
A6: 1-1 < 1-pp by XREAL_1:15;
  then
A7: 0 < q by A2;
A8: 0 <= b to_power q / q
  proof
    now
      per cases by A4;
      case
        0 < b;
        then 0 < b to_power q by POWER:34;
        hence thesis by A7;
      end;
      case
        0 = b;
        then 0 = b to_power q by A7,POWER:def 2;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  now
    per cases;
    case
A9:   a*b=0;
      now
        per cases by A9,XCMPLX_1:6;
        case
A10:      a=0;
A11:      now
            assume a * b = a to_power p / p + b to_power q / q;
            then 0 = 0/p + b to_power q / q by A1,A10,POWER:def 2;
            then 0*q = (b to_power q /q)*q;
            then 0 = b to_power q by A7,XCMPLX_1:87;
            then
A12:        0 = (b to_power q ) to_power (1/q) by A2,A6,POWER:def 2;
A13:        0=b
            proof
              assume
A14:          b <> 0;
              then 0 = b to_power (q * (1/q) ) by A4,A12,POWER:33;
              then 0 = b to_power 1 by A7,XCMPLX_1:106;
              hence contradiction by A14,POWER:25;
            end;
            thus a to_power p = 0 by A1,A10,POWER:def 2
              .= b to_power q by A7,A13,POWER:def 2;
          end;
          now
            assume a to_power p = b to_power q;
            then
A15:        b to_power q /q =0 /q by A1,A10,POWER:def 2
              .=0;
            a to_power p /p = 0/p by A1,A10,POWER:def 2
              .=0;
            hence a*b=a to_power p /p + b to_power q /q by A9,A15;
          end;
          hence a * b = a to_power p / p + b to_power q / q iff a to_power p =
          b to_power q by A11;
        end;
        case
A16:      b=0;
A17:      1/p > 0 by A1,XREAL_1:139;
A18:      now
            assume a * b = a to_power p / p + b to_power q / q;
            then 0 = 0/q + a to_power p / p by A7,A16,POWER:def 2;
            then 0*p = (a to_power p/p)*p;
            then 0 = a to_power p by A1,XCMPLX_1:87;
            then
A19:        0 = (a to_power p ) to_power (1/p) by A17,POWER:def 2;
A20:        0=a
            proof
              assume
A21:          a <> 0;
              then 0 = a to_power (p * (1/p) ) by A3,A19,POWER:33;
              then 0 = a to_power 1 by A1,XCMPLX_1:106;
              hence contradiction by A21,POWER:25;
            end;
            thus b to_power q = 0 by A7,A16,POWER:def 2
              .= a to_power p by A1,A20,POWER:def 2;
          end;
          now
            assume a to_power p = b to_power q;
            then
A22:        a to_power p /p =0 /p by A7,A16,POWER:def 2
              .=0;
            b to_power q /q = 0/q by A7,A16,POWER:def 2
              .= 0;
            hence a*b=a to_power p /p + b to_power q /q by A9,A22;
          end;
          hence a * b = a to_power p / p + b to_power q / q iff a to_power p =
          b to_power q by A18;
        end;
      end;
      hence thesis by A5,A8;
    end;
    case
A23:  a*b<>0;
      then
A24:  a <> 0;
A25:  b <> 0 by A23;
      then
      a * b = a #R p / p + b #R q / q iff a #R p = b #R q by A1,A2,A3,A4,A24
,Th4;
      then
A26:  a * b = a to_power p / p + b #R q / q iff a to_power p = b #R q by A3,A24
,POWER:def 2;
      a * b <= a #R p / p + b #R q / q by A1,A2,A3,A4,A24,A25,Th4;
      then a * b <= a to_power p / p + b #R q / q by A3,A24,POWER:def 2;
      hence thesis by A4,A25,A26,POWER:def 2;
    end;
  end;
  hence thesis;
end;
