let p be Prime; :: thesis:
assume p < 773 ; :: thesis:
then ( 1 + 1 < p + 1 & p < 772 + 1 ) by XREAL_1:6, INT_2:def 4;

suppose ( 2 <= p & p < 769 ) ; :: thesis:

hence ( p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 ) by Ttool769a; :: thesis:

end;

suppose ( ( 769 <= p & p <= 769 + 1 ) or ( 770 <= p & p <= 770 + 1 ) or ( 771 <= p & p <= 771 + 1 ) ) ; :: thesis:

then p = 769 by XPRIMES0:770, XPRIMES0:771, XPRIMES0:772, NAT_1:9;
hence ( p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 ) ; :: thesis:

end;