:: GEOMTRAP semantic presentation begin definition let V be RealLinearSpace; let u, v, u1, v1 be VECTOR of V; predu,v '||' u1,v1 means :Def1: :: GEOMTRAP:def 1 ( u,v // u1,v1 or u,v // v1,u1 ); end; :: deftheorem Def1 defines '||' GEOMTRAP:def_1_:_ for V being RealLinearSpace for u, v, u1, v1 being VECTOR of V holds ( u,v '||' u1,v1 iff ( u,v // u1,v1 or u,v // v1,u1 ) ); theorem Th1: :: GEOMTRAP:1 for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds OASpace V is OAffinSpace proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V st Gen w,y holds OASpace V is OAffinSpace let w, y be VECTOR of V; ::_thesis: ( Gen w,y implies OASpace V is OAffinSpace ) assume Gen w,y ; ::_thesis: OASpace V is OAffinSpace then for a1, a2 being Real st (a1 * w) + (a2 * y) = 0. V holds ( a1 = 0 & a2 = 0 ) by ANALMETR:def_1; hence OASpace V is OAffinSpace by ANALOAF:26; ::_thesis: verum end; theorem Th2: :: GEOMTRAP:2 for V being RealLinearSpace for u, v, u1, v1 being VECTOR of V for p, q, p1, q1 being Element of (OASpace V) st p = u & q = v & p1 = u1 & q1 = v1 holds ( p,q // p1,q1 iff u,v // u1,v1 ) proof let V be RealLinearSpace; ::_thesis: for u, v, u1, v1 being VECTOR of V for p, q, p1, q1 being Element of (OASpace V) st p = u & q = v & p1 = u1 & q1 = v1 holds ( p,q // p1,q1 iff u,v // u1,v1 ) let u, v, u1, v1 be VECTOR of V; ::_thesis: for p, q, p1, q1 being Element of (OASpace V) st p = u & q = v & p1 = u1 & q1 = v1 holds ( p,q // p1,q1 iff u,v // u1,v1 ) A1: OASpace V = AffinStruct(# the carrier of V,(DirPar V) #) by ANALOAF:def_4; let p, q, p1, q1 be Element of (OASpace V); ::_thesis: ( p = u & q = v & p1 = u1 & q1 = v1 implies ( p,q // p1,q1 iff u,v // u1,v1 ) ) assume A2: ( p = u & q = v & p1 = u1 & q1 = v1 ) ; ::_thesis: ( p,q // p1,q1 iff u,v // u1,v1 ) A3: now__::_thesis:_(_u,v_//_u1,v1_implies_p,q_//_p1,q1_) assume u,v // u1,v1 ; ::_thesis: p,q // p1,q1 then [[p,q],[p1,q1]] in the CONGR of (OASpace V) by A2, A1, ANALOAF:22; hence p,q // p1,q1 by ANALOAF:def_2; ::_thesis: verum end; now__::_thesis:_(_p,q_//_p1,q1_implies_u,v_//_u1,v1_) assume p,q // p1,q1 ; ::_thesis: u,v // u1,v1 then [[u,v],[u1,v1]] in DirPar V by A2, A1, ANALOAF:def_2; hence u,v // u1,v1 by ANALOAF:22; ::_thesis: verum end; hence ( p,q // p1,q1 iff u,v // u1,v1 ) by A3; ::_thesis: verum end; theorem Th3: :: GEOMTRAP:3 for V being RealLinearSpace for w, y, u, v, u1, v1 being VECTOR of V st Gen w,y holds for p, q, p1, q1 being Element of the carrier of (Lambda (OASpace V)) st p = u & q = v & p1 = u1 & q1 = v1 holds ( p,q // p1,q1 iff u,v '||' u1,v1 ) proof let V be RealLinearSpace; ::_thesis: for w, y, u, v, u1, v1 being VECTOR of V st Gen w,y holds for p, q, p1, q1 being Element of the carrier of (Lambda (OASpace V)) st p = u & q = v & p1 = u1 & q1 = v1 holds ( p,q // p1,q1 iff u,v '||' u1,v1 ) let w, y, u, v, u1, v1 be VECTOR of V; ::_thesis: ( Gen w,y implies for p, q, p1, q1 being Element of the carrier of (Lambda (OASpace V)) st p = u & q = v & p1 = u1 & q1 = v1 holds ( p,q // p1,q1 iff u,v '||' u1,v1 ) ) assume Gen w,y ; ::_thesis: for p, q, p1, q1 being Element of the carrier of (Lambda (OASpace V)) st p = u & q = v & p1 = u1 & q1 = v1 holds ( p,q // p1,q1 iff u,v '||' u1,v1 ) then reconsider S = OASpace V as OAffinSpace by Th1; let p, q, p1, q1 be Element of the carrier of (Lambda (OASpace V)); ::_thesis: ( p = u & q = v & p1 = u1 & q1 = v1 implies ( p,q // p1,q1 iff u,v '||' u1,v1 ) ) assume A1: ( p = u & q = v & p1 = u1 & q1 = v1 ) ; ::_thesis: ( p,q // p1,q1 iff u,v '||' u1,v1 ) Lambda (OASpace V) = AffinStruct(# the carrier of (OASpace V),(lambda the CONGR of (OASpace V)) #) by DIRAF:def_2; then reconsider p9 = p, q9 = q, p19 = p1, q19 = q1 as Element of S ; A2: now__::_thesis:_(_u,v_'||'_u1,v1_implies_p,q_//_p1,q1_) assume u,v '||' u1,v1 ; ::_thesis: p,q // p1,q1 then ( u,v // u1,v1 or u,v // v1,u1 ) by Def1; then ( p9,q9 // p19,q19 or p9,q9 // q19,p19 ) by A1, Th2; then p9,q9 '||' p19,q19 by DIRAF:def_4; hence p,q // p1,q1 by DIRAF:38; ::_thesis: verum end; now__::_thesis:_(_p,q_//_p1,q1_implies_u,v_'||'_u1,v1_) assume p,q // p1,q1 ; ::_thesis: u,v '||' u1,v1 then p9,q9 '||' p19,q19 by DIRAF:38; then ( p9,q9 // p19,q19 or p9,q9 // q19,p19 ) by DIRAF:def_4; then ( u,v // u1,v1 or u,v // v1,u1 ) by A1, Th2; hence u,v '||' u1,v1 by Def1; ::_thesis: verum end; hence ( p,q // p1,q1 iff u,v '||' u1,v1 ) by A2; ::_thesis: verum end; theorem Th4: :: GEOMTRAP:4 for V being RealLinearSpace for w, y, u, v, u1, v1 being VECTOR of V for p, q, p1, q1 being Element of the carrier of (AMSpace (V,w,y)) st p = u & q = v & p1 = u1 & q1 = v1 holds ( p,q // p1,q1 iff u,v '||' u1,v1 ) proof let V be RealLinearSpace; ::_thesis: for w, y, u, v, u1, v1 being VECTOR of V for p, q, p1, q1 being Element of the carrier of (AMSpace (V,w,y)) st p = u & q = v & p1 = u1 & q1 = v1 holds ( p,q // p1,q1 iff u,v '||' u1,v1 ) let w, y, u, v, u1, v1 be VECTOR of V; ::_thesis: for p, q, p1, q1 being Element of the carrier of (AMSpace (V,w,y)) st p = u & q = v & p1 = u1 & q1 = v1 holds ( p,q // p1,q1 iff u,v '||' u1,v1 ) let p, q, p1, q1 be Element of the carrier of (AMSpace (V,w,y)); ::_thesis: ( p = u & q = v & p1 = u1 & q1 = v1 implies ( p,q // p1,q1 iff u,v '||' u1,v1 ) ) assume A1: ( p = u & q = v & p1 = u1 & q1 = v1 ) ; ::_thesis: ( p,q // p1,q1 iff u,v '||' u1,v1 ) A2: now__::_thesis:_(_p,q_//_p1,q1_implies_u,v_'||'_u1,v1_) assume p,q // p1,q1 ; ::_thesis: u,v '||' u1,v1 then ex a, b being Real st ( a * (v - u) = b * (v1 - u1) & ( a <> 0 or b <> 0 ) ) by A1, ANALMETR:22; then ( u,v // u1,v1 or u,v // v1,u1 ) by ANALMETR:14; hence u,v '||' u1,v1 by Def1; ::_thesis: verum end; now__::_thesis:_(_u,v_'||'_u1,v1_implies_p,q_//_p1,q1_) assume u,v '||' u1,v1 ; ::_thesis: p,q // p1,q1 then ( u,v // u1,v1 or u,v // v1,u1 ) by Def1; then ex a, b being Real st ( a * (v - u) = b * (v1 - u1) & ( a <> 0 or b <> 0 ) ) by ANALMETR:14; hence p,q // p1,q1 by A1, ANALMETR:22; ::_thesis: verum end; hence ( p,q // p1,q1 iff u,v '||' u1,v1 ) by A2; ::_thesis: verum end; definition let V be RealLinearSpace; let u, v be VECTOR of V; funcu # v -> VECTOR of V means :Def2: :: GEOMTRAP:def 2 it + it = u + v; existence ex b1 being VECTOR of V st b1 + b1 = u + v proof set y = u + v; set w = (2 ") * (u + v); (2 ") + (2 ") = 1 ; then ((2 ") * (u + v)) + ((2 ") * (u + v)) = 1 * (u + v) by RLVECT_1:def_6 .= u + v by RLVECT_1:def_8 ; hence ex b1 being VECTOR of V st b1 + b1 = u + v ; ::_thesis: verum end; uniqueness for b1, b2 being VECTOR of V st b1 + b1 = u + v & b2 + b2 = u + v holds b1 = b2 proof let w, w1 be VECTOR of V; ::_thesis: ( w + w = u + v & w1 + w1 = u + v implies w = w1 ) assume ( w + w = u + v & w1 + w1 = u + v ) ; ::_thesis: w = w1 then 0. V = (w + w) - (w1 + w1) by RLVECT_1:15 .= w + (w - (w1 + w1)) by RLVECT_1:def_3 .= w + ((w - w1) - w1) by RLVECT_1:27 .= (w + (w - w1)) - w1 by RLVECT_1:def_3 .= (w - w1) + (w - w1) by RLVECT_1:def_3 ; then w - w1 = 0. V by RLVECT_1:20; hence w = w1 by RLVECT_1:21; ::_thesis: verum end; commutativity for b1, u, v being VECTOR of V st b1 + b1 = u + v holds b1 + b1 = v + u ; idempotence for u being VECTOR of V holds u + u = u + u ; end; :: deftheorem Def2 defines # GEOMTRAP:def_2_:_ for V being RealLinearSpace for u, v, b4 being VECTOR of V holds ( b4 = u # v iff b4 + b4 = u + v ); theorem Th5: :: GEOMTRAP:5 for V being RealLinearSpace for u, w being VECTOR of V ex y being VECTOR of V st u # y = w proof let V be RealLinearSpace; ::_thesis: for u, w being VECTOR of V ex y being VECTOR of V st u # y = w let u, w be VECTOR of V; ::_thesis: ex y being VECTOR of V st u # y = w take y = (- u) + (w + w); ::_thesis: u # y = w u + y = (u + (- u)) + (w + w) by RLVECT_1:def_3 .= (0. V) + (w + w) by RLVECT_1:5 .= w + w by RLVECT_1:4 ; hence u # y = w by Def2; ::_thesis: verum end; theorem Th6: :: GEOMTRAP:6 for V being RealLinearSpace for u, u1, v, v1 being VECTOR of V holds (u # u1) # (v # v1) = (u # v) # (u1 # v1) proof let V be RealLinearSpace; ::_thesis: for u, u1, v, v1 being VECTOR of V holds (u # u1) # (v # v1) = (u # v) # (u1 # v1) let u, u1, v, v1 be VECTOR of V; ::_thesis: (u # u1) # (v # v1) = (u # v) # (u1 # v1) set p = u # u1; set q = v # v1; set r = u # v; set s = u1 # v1; set x = (u # u1) # (v # v1); set y = (u # v) # (u1 # v1); A1: now__::_thesis:_for_w_being_VECTOR_of_V_holds_(w_+_w)_+_(w_+_w)_=_4_*_w let w be VECTOR of V; ::_thesis: (w + w) + (w + w) = 4 * w w = 1 * w by RLVECT_1:def_8; then (w + w) + (w + w) = ((1 + 1) * w) + ((1 * w) + (1 * w)) by RLVECT_1:def_6 .= ((1 + 1) * w) + ((1 + 1) * w) by RLVECT_1:def_6 .= ((1 + 1) + (1 + 1)) * w by RLVECT_1:def_6 ; hence (w + w) + (w + w) = 4 * w ; ::_thesis: verum end; (((u # u1) # (v # v1)) + ((u # u1) # (v # v1))) + (((u # u1) # (v # v1)) + ((u # u1) # (v # v1))) = (((u # u1) # (v # v1)) + ((u # u1) # (v # v1))) + ((u # u1) + (v # v1)) by Def2 .= ((u # u1) + (v # v1)) + ((u # u1) + (v # v1)) by Def2 .= (u # u1) + ((v # v1) + ((u # u1) + (v # v1))) by RLVECT_1:def_3 .= (u # u1) + ((u # u1) + ((v # v1) + (v # v1))) by RLVECT_1:def_3 .= ((u # u1) + (u # u1)) + ((v # v1) + (v # v1)) by RLVECT_1:def_3 .= ((u # u1) + (u # u1)) + (v + v1) by Def2 .= (u + u1) + (v + v1) by Def2 .= u + (u1 + (v + v1)) by RLVECT_1:def_3 .= u + (v + (v1 + u1)) by RLVECT_1:def_3 .= (u + v) + (v1 + u1) by RLVECT_1:def_3 .= (u + v) + ((u1 # v1) + (u1 # v1)) by Def2 .= ((u # v) + (u # v)) + ((u1 # v1) + (u1 # v1)) by Def2 .= (u # v) + ((u # v) + ((u1 # v1) + (u1 # v1))) by RLVECT_1:def_3 .= (u # v) + ((u1 # v1) + ((u1 # v1) + (u # v))) by RLVECT_1:def_3 .= ((u # v) + (u1 # v1)) + ((u1 # v1) + (u # v)) by RLVECT_1:def_3 .= (((u # v) # (u1 # v1)) + ((u # v) # (u1 # v1))) + ((u # v) + (u1 # v1)) by Def2 .= (((u # v) # (u1 # v1)) + ((u # v) # (u1 # v1))) + (((u # v) # (u1 # v1)) + ((u # v) # (u1 # v1))) by Def2 ; then 4 * ((u # u1) # (v # v1)) = (((u # v) # (u1 # v1)) + ((u # v) # (u1 # v1))) + (((u # v) # (u1 # v1)) + ((u # v) # (u1 # v1))) by A1 .= 4 * ((u # v) # (u1 # v1)) by A1 ; hence (u # u1) # (v # v1) = (u # v) # (u1 # v1) by RLVECT_1:36; ::_thesis: verum end; theorem Th7: :: GEOMTRAP:7 for V being RealLinearSpace for u, y, w being VECTOR of V st u # y = u # w holds y = w proof let V be RealLinearSpace; ::_thesis: for u, y, w being VECTOR of V st u # y = u # w holds y = w let u, y, w be VECTOR of V; ::_thesis: ( u # y = u # w implies y = w ) assume A1: u # y = u # w ; ::_thesis: y = w set p = u # y; u + y = (u # y) + (u # y) by Def2 .= u + w by A1, Def2 ; hence y = w by RLVECT_1:8; ::_thesis: verum end; theorem Th8: :: GEOMTRAP:8 for V being RealLinearSpace for u, v, y being VECTOR of V holds u,v // y # u,y # v proof let V be RealLinearSpace; ::_thesis: for u, v, y being VECTOR of V holds u,v // y # u,y # v let u, v, y be VECTOR of V; ::_thesis: u,v // y # u,y # v set p = y # u; set r = y # v; A1: ( y + u = (y # u) + (y # u) & y + v = (y # v) + (y # v) ) by Def2; 2 * ((y # v) - (y # u)) = ((1 + 1) * (y # v)) - ((1 + 1) * (y # u)) by RLVECT_1:34 .= ((1 * (y # v)) + (1 * (y # v))) - ((1 + 1) * (y # u)) by RLVECT_1:def_6 .= ((1 * (y # v)) + (1 * (y # v))) - ((1 * (y # u)) + (1 * (y # u))) by RLVECT_1:def_6 .= ((y # v) + (1 * (y # v))) - ((1 * (y # u)) + (1 * (y # u))) by RLVECT_1:def_8 .= ((y # v) + (y # v)) - ((1 * (y # u)) + (1 * (y # u))) by RLVECT_1:def_8 .= ((y # v) + (y # v)) - ((y # u) + (1 * (y # u))) by RLVECT_1:def_8 .= (y + v) - (y + u) by A1, RLVECT_1:def_8 .= v + (y - (y + u)) by RLVECT_1:def_3 .= v + ((y - y) - u) by RLVECT_1:27 .= v + ((0. V) - u) by RLVECT_1:15 .= v - u by RLVECT_1:14 .= 1 * (v - u) by RLVECT_1:def_8 ; hence u,v // y # u,y # v by ANALOAF:def_1; ::_thesis: verum end; theorem :: GEOMTRAP:9 for V being RealLinearSpace for u, v, w being VECTOR of V holds u,v '||' w # u,w # v proof let V be RealLinearSpace; ::_thesis: for u, v, w being VECTOR of V holds u,v '||' w # u,w # v let u, v, w be VECTOR of V; ::_thesis: u,v '||' w # u,w # v u,v // w # u,w # v by Th8; hence u,v '||' w # u,w # v by Def1; ::_thesis: verum end; theorem Th10: :: GEOMTRAP:10 for V being RealLinearSpace for u, v being VECTOR of V holds ( 2 * ((u # v) - u) = v - u & 2 * (v - (u # v)) = v - u ) proof let V be RealLinearSpace; ::_thesis: for u, v being VECTOR of V holds ( 2 * ((u # v) - u) = v - u & 2 * (v - (u # v)) = v - u ) let u, v be VECTOR of V; ::_thesis: ( 2 * ((u # v) - u) = v - u & 2 * (v - (u # v)) = v - u ) set p = u # v; A1: 2 - 1 = 1 ; A2: 2 * (v - (u # v)) = (2 * v) - ((1 + 1) * (u # v)) by RLVECT_1:34 .= (2 * v) - ((1 * (u # v)) + (1 * (u # v))) by RLVECT_1:def_6 .= (2 * v) - ((1 * (u # v)) + (u # v)) by RLVECT_1:def_8 .= (2 * v) - ((u # v) + (u # v)) by RLVECT_1:def_8 .= (2 * v) - (u + v) by Def2 .= ((2 * v) - v) - u by RLVECT_1:27 .= ((2 * v) - (1 * v)) - u by RLVECT_1:def_8 .= (1 * v) - u by A1, RLVECT_1:35 .= v - u by RLVECT_1:def_8 ; A3: 1 - 2 = - 1 ; 2 * ((u # v) - u) = ((1 + 1) * (u # v)) - (2 * u) by RLVECT_1:34 .= ((1 * (u # v)) + (1 * (u # v))) - (2 * u) by RLVECT_1:def_6 .= ((u # v) + (1 * (u # v))) - (2 * u) by RLVECT_1:def_8 .= ((u # v) + (u # v)) - (2 * u) by RLVECT_1:def_8 .= (u + v) - (2 * u) by Def2 .= v + (u - (2 * u)) by RLVECT_1:def_3 .= v + ((1 * u) - (2 * u)) by RLVECT_1:def_8 .= v + ((- 1) * u) by A3, RLVECT_1:35 .= v - u by RLVECT_1:16 ; hence ( 2 * ((u # v) - u) = v - u & 2 * (v - (u # v)) = v - u ) by A2; ::_thesis: verum end; theorem Th11: :: GEOMTRAP:11 for V being RealLinearSpace for u, v being VECTOR of V holds u,u # v // u # v,v proof let V be RealLinearSpace; ::_thesis: for u, v being VECTOR of V holds u,u # v // u # v,v let u, v be VECTOR of V; ::_thesis: u,u # v // u # v,v set p = u # v; 2 * ((u # v) - u) = v - u by Th10 .= 2 * (v - (u # v)) by Th10 ; hence u,u # v // u # v,v by ANALOAF:def_1; ::_thesis: verum end; theorem Th12: :: GEOMTRAP:12 for V being RealLinearSpace for u, v being VECTOR of V holds ( u,v // u,u # v & u,v // u # v,v ) proof let V be RealLinearSpace; ::_thesis: for u, v being VECTOR of V holds ( u,v // u,u # v & u,v // u # v,v ) let u, v be VECTOR of V; ::_thesis: ( u,v // u,u # v & u,v // u # v,v ) set p = u # v; 1 * (v - u) = v - u by RLVECT_1:def_8 .= 2 * ((u # v) - u) by Th10 ; hence u,v // u,u # v by ANALOAF:def_1; ::_thesis: u,v // u # v,v 1 * (v - u) = v - u by RLVECT_1:def_8 .= 2 * (v - (u # v)) by Th10 ; hence u,v // u # v,v by ANALOAF:def_1; ::_thesis: verum end; Lm1: for V being RealLinearSpace for u, y, v being VECTOR of V st u,y // y,v holds ( v,y // y,u & u,y // u,v & y,v // u,v ) proof let V be RealLinearSpace; ::_thesis: for u, y, v being VECTOR of V st u,y // y,v holds ( v,y // y,u & u,y // u,v & y,v // u,v ) let u, y, v be VECTOR of V; ::_thesis: ( u,y // y,v implies ( v,y // y,u & u,y // u,v & y,v // u,v ) ) assume A1: u,y // y,v ; ::_thesis: ( v,y // y,u & u,y // u,v & y,v // u,v ) then y,u // v,y by ANALOAF:12; hence A2: v,y // y,u by ANALOAF:12; ::_thesis: ( u,y // u,v & y,v // u,v ) thus u,y // u,v by A1, ANALOAF:13; ::_thesis: y,v // u,v v,y // v,u by A2, ANALOAF:13; hence y,v // u,v by ANALOAF:12; ::_thesis: verum end; theorem Th13: :: GEOMTRAP:13 for V being RealLinearSpace for u, y, v being VECTOR of V st u,y // y,v holds u # y,y // y,y # v proof let V be RealLinearSpace; ::_thesis: for u, y, v being VECTOR of V st u,y // y,v holds u # y,y // y,y # v let u, y, v be VECTOR of V; ::_thesis: ( u,y // y,v implies u # y,y // y,y # v ) set p = u # y; set q = y # v; u,u # y // u # y,y by Th11; then u # y,y // u,y by Lm1; then A1: u,y // u # y,y by ANALOAF:12; y,y # v // y # v,v by Th11; then y,y # v // y,v by Lm1; then A2: y,v // y,y # v by ANALOAF:12; assume A3: u,y // y,v ; ::_thesis: u # y,y // y,y # v now__::_thesis:_(_u_<>_y_&_y_<>_v_implies_u_#_y,y_//_y,y_#_v_) assume that A4: u <> y and A5: y <> v ; ::_thesis: u # y,y // y,y # v y,v // u # y,y by A3, A1, A4, ANALOAF:11; hence u # y,y // y,y # v by A2, A5, ANALOAF:11; ::_thesis: verum end; hence u # y,y // y,y # v by ANALOAF:9; ::_thesis: verum end; theorem Th14: :: GEOMTRAP:14 for V being RealLinearSpace for u, v, u1, v1 being VECTOR of V st u,v // u1,v1 holds u,v // u # u1,v # v1 proof let V be RealLinearSpace; ::_thesis: for u, v, u1, v1 being VECTOR of V st u,v // u1,v1 holds u,v // u # u1,v # v1 let u, v, u1, v1 be VECTOR of V; ::_thesis: ( u,v // u1,v1 implies u,v // u # u1,v # v1 ) assume A1: u,v // u1,v1 ; ::_thesis: u,v // u # u1,v # v1 percases ( u = v or u1 = v1 or ( u <> v & u1 <> v1 ) ) ; suppose ( u = v or u1 = v1 ) ; ::_thesis: u,v // u # u1,v # v1 hence u,v // u # u1,v # v1 by Th8, ANALOAF:9; ::_thesis: verum end; supposeA2: ( u <> v & u1 <> v1 ) ; ::_thesis: u,v // u # u1,v # v1 set p = u # u1; set q = v # v1; consider a, b being Real such that A3: ( 0 < a & 0 < b ) and A4: a * (v - u) = b * (v1 - u1) by A1, A2, ANALOAF:def_1; A5: ( 0 < a + b & 0 < b * 2 ) by A3, XREAL_1:34, XREAL_1:129; (b * 2) * ((v # v1) - (u # u1)) = b * (2 * ((v # v1) - (u # u1))) by RLVECT_1:def_7 .= b * (((1 + 1) * (v # v1)) - (2 * (u # u1))) by RLVECT_1:34 .= b * (((1 * (v # v1)) + (1 * (v # v1))) - (2 * (u # u1))) by RLVECT_1:def_6 .= b * (((v # v1) + (1 * (v # v1))) - (2 * (u # u1))) by RLVECT_1:def_8 .= b * (((v # v1) + (v # v1)) - (2 * (u # u1))) by RLVECT_1:def_8 .= b * ((v + v1) - (2 * (u # u1))) by Def2 .= b * (v + (v1 - ((1 + 1) * (u # u1)))) by RLVECT_1:def_3 .= b * (v + (v1 - ((1 * (u # u1)) + (1 * (u # u1))))) by RLVECT_1:def_6 .= b * (v + (v1 - ((u # u1) + (1 * (u # u1))))) by RLVECT_1:def_8 .= b * (v + (v1 - ((u # u1) + (u # u1)))) by RLVECT_1:def_8 .= b * (v + (v1 - (u + u1))) by Def2 .= b * (v + ((v1 - u1) - u)) by RLVECT_1:27 .= b * ((v + (v1 - u1)) - u) by RLVECT_1:def_3 .= b * ((v1 - u1) + (v - u)) by RLVECT_1:def_3 .= (a * (v - u)) + (b * (v - u)) by A4, RLVECT_1:def_5 .= (a + b) * (v - u) by RLVECT_1:def_6 ; hence u,v // u # u1,v # v1 by A5, ANALOAF:def_1; ::_thesis: verum end; end; end; Lm2: for V being RealLinearSpace for u, v, u1, v1 being VECTOR of V st u,v // u1,v1 holds u,v '||' u # v1,v # u1 proof let V be RealLinearSpace; ::_thesis: for u, v, u1, v1 being VECTOR of V st u,v // u1,v1 holds u,v '||' u # v1,v # u1 let u, v, u1, v1 be VECTOR of V; ::_thesis: ( u,v // u1,v1 implies u,v '||' u # v1,v # u1 ) assume A1: u,v // u1,v1 ; ::_thesis: u,v '||' u # v1,v # u1 percases ( u = v or u1 = v1 or ( u <> v & u1 <> v1 ) ) ; suppose u = v ; ::_thesis: u,v '||' u # v1,v # u1 then u,v // u # v1,v # u1 by ANALOAF:9; hence u,v '||' u # v1,v # u1 by Def1; ::_thesis: verum end; suppose u1 = v1 ; ::_thesis: u,v '||' u # v1,v # u1 then u,v // u # v1,v # u1 by Th8; hence u,v '||' u # v1,v # u1 by Def1; ::_thesis: verum end; suppose ( u <> v & u1 <> v1 ) ; ::_thesis: u,v '||' u # v1,v # u1 then consider a, b being Real such that 0 < a and A2: 0 < b and A3: a * (v - u) = b * (v1 - u1) by A1, ANALOAF:def_1; A4: b * (u1 - v1) = b * (- (v1 - u1)) by RLVECT_1:33 .= - (a * (v - u)) by A3, RLVECT_1:25 .= a * (- (v - u)) by RLVECT_1:25 .= (- a) * (v - u) by RLVECT_1:24 ; set p = u # v1; set q = v # u1; set A = b * 2; set D = b - a; A5: b * 2 <> 0 by A2; (b * 2) * ((v # u1) - (u # v1)) = b * (2 * ((v # u1) - (u # v1))) by RLVECT_1:def_7 .= b * (((1 + 1) * (v # u1)) - (2 * (u # v1))) by RLVECT_1:34 .= b * (((1 * (v # u1)) + (1 * (v # u1))) - (2 * (u # v1))) by RLVECT_1:def_6 .= b * (((v # u1) + (1 * (v # u1))) - (2 * (u # v1))) by RLVECT_1:def_8 .= b * (((v # u1) + (v # u1)) - (2 * (u # v1))) by RLVECT_1:def_8 .= b * ((v + u1) - (2 * (u # v1))) by Def2 .= b * (v + (u1 - ((1 + 1) * (u # v1)))) by RLVECT_1:def_3 .= b * (v + (u1 - ((1 * (u # v1)) + (1 * (u # v1))))) by RLVECT_1:def_6 .= b * (v + (u1 - ((u # v1) + (1 * (u # v1))))) by RLVECT_1:def_8 .= b * (v + (u1 - ((u # v1) + (u # v1)))) by RLVECT_1:def_8 .= b * (v + (u1 - (u + v1))) by Def2 .= b * (v + ((u1 - v1) - u)) by RLVECT_1:27 .= b * ((v + (u1 - v1)) - u) by RLVECT_1:def_3 .= b * ((u1 - v1) + (v - u)) by RLVECT_1:def_3 .= ((- a) * (v - u)) + (b * (v - u)) by A4, RLVECT_1:def_5 .= (b + (- a)) * (v - u) by RLVECT_1:def_6 .= (b - a) * (v - u) ; then ( u,v // u # v1,v # u1 or u,v // v # u1,u # v1 ) by A5, ANALMETR:14; hence u,v '||' u # v1,v # u1 by Def1; ::_thesis: verum end; end; end; Lm3: for V being RealLinearSpace for u1, u2, v1, v2 being VECTOR of V st u1 # u2 = v1 # v2 holds v1 - u1 = - (v2 - u2) proof let V be RealLinearSpace; ::_thesis: for u1, u2, v1, v2 being VECTOR of V st u1 # u2 = v1 # v2 holds v1 - u1 = - (v2 - u2) let u1, u2, v1, v2 be VECTOR of V; ::_thesis: ( u1 # u2 = v1 # v2 implies v1 - u1 = - (v2 - u2) ) set p = u1 # u2; A1: (u1 # u2) + (u1 # u2) = u1 + u2 by Def2; assume u1 # u2 = v1 # v2 ; ::_thesis: v1 - u1 = - (v2 - u2) then A2: (u1 # u2) + (u1 # u2) = v1 + v2 by Def2; (v1 - u1) + v2 = (v2 + v1) - u1 by RLVECT_1:def_3 .= u2 by A1, A2, RLSUB_2:61 ; then (v1 - u1) + (v2 - u2) = u2 - u2 by RLVECT_1:def_3 .= 0. V by RLVECT_1:15 ; hence v1 - u1 = - (v2 - u2) by RLVECT_1:6; ::_thesis: verum end; Lm4: for V being RealLinearSpace for u, v, u1, v1, w, y being VECTOR of V st u,v,u1,v1 are_Ort_wrt w,y holds u,v,v1,u1 are_Ort_wrt w,y proof let V be RealLinearSpace; ::_thesis: for u, v, u1, v1, w, y being VECTOR of V st u,v,u1,v1 are_Ort_wrt w,y holds u,v,v1,u1 are_Ort_wrt w,y let u, v, u1, v1, w, y be VECTOR of V; ::_thesis: ( u,v,u1,v1 are_Ort_wrt w,y implies u,v,v1,u1 are_Ort_wrt w,y ) assume u,v,u1,v1 are_Ort_wrt w,y ; ::_thesis: u,v,v1,u1 are_Ort_wrt w,y then v - u,v1 - u1 are_Ort_wrt w,y by ANALMETR:def_3; then A1: v - u,(- 1) * (v1 - u1) are_Ort_wrt w,y by ANALMETR:7; (- 1) * (v1 - u1) = - (v1 - u1) by RLVECT_1:16 .= u1 - v1 by RLVECT_1:33 ; hence u,v,v1,u1 are_Ort_wrt w,y by A1, ANALMETR:def_3; ::_thesis: verum end; Lm5: for V being RealLinearSpace for u, v, u1, v1, w, y being VECTOR of V st u,v,u1,v1 are_Ort_wrt w,y holds u1,v1,u,v are_Ort_wrt w,y proof let V be RealLinearSpace; ::_thesis: for u, v, u1, v1, w, y being VECTOR of V st u,v,u1,v1 are_Ort_wrt w,y holds u1,v1,u,v are_Ort_wrt w,y let u, v, u1, v1, w, y be VECTOR of V; ::_thesis: ( u,v,u1,v1 are_Ort_wrt w,y implies u1,v1,u,v are_Ort_wrt w,y ) assume u,v,u1,v1 are_Ort_wrt w,y ; ::_thesis: u1,v1,u,v are_Ort_wrt w,y then v - u,v1 - u1 are_Ort_wrt w,y by ANALMETR:def_3; then v1 - u1,v - u are_Ort_wrt w,y by ANALMETR:4; hence u1,v1,u,v are_Ort_wrt w,y by ANALMETR:def_3; ::_thesis: verum end; Lm6: for V being RealLinearSpace for w, y, u, v, u1 being VECTOR of V st Gen w,y holds u,v,u1,u1 are_Ort_wrt w,y proof let V be RealLinearSpace; ::_thesis: for w, y, u, v, u1 being VECTOR of V st Gen w,y holds u,v,u1,u1 are_Ort_wrt w,y let w, y, u, v, u1 be VECTOR of V; ::_thesis: ( Gen w,y implies u,v,u1,u1 are_Ort_wrt w,y ) A1: u1 - u1 = 0. V by RLVECT_1:15; assume Gen w,y ; ::_thesis: u,v,u1,u1 are_Ort_wrt w,y then v - u,u1 - u1 are_Ort_wrt w,y by A1, ANALMETR:5; hence u,v,u1,u1 are_Ort_wrt w,y by ANALMETR:def_3; ::_thesis: verum end; Lm7: for V being RealLinearSpace for w, y, u, v, w1, v1, v2 being VECTOR of V st Gen w,y & u,v,w1,v1 are_Ort_wrt w,y & u,v,w1,v2 are_Ort_wrt w,y holds u,v,v1,v2 are_Ort_wrt w,y proof let V be RealLinearSpace; ::_thesis: for w, y, u, v, w1, v1, v2 being VECTOR of V st Gen w,y & u,v,w1,v1 are_Ort_wrt w,y & u,v,w1,v2 are_Ort_wrt w,y holds u,v,v1,v2 are_Ort_wrt w,y let w, y, u, v, w1, v1, v2 be VECTOR of V; ::_thesis: ( Gen w,y & u,v,w1,v1 are_Ort_wrt w,y & u,v,w1,v2 are_Ort_wrt w,y implies u,v,v1,v2 are_Ort_wrt w,y ) assume that A1: Gen w,y and A2: ( u,v,w1,v1 are_Ort_wrt w,y & u,v,w1,v2 are_Ort_wrt w,y ) ; ::_thesis: u,v,v1,v2 are_Ort_wrt w,y ( v - u,v1 - w1 are_Ort_wrt w,y & v - u,v2 - w1 are_Ort_wrt w,y ) by A2, ANALMETR:def_3; then A3: v - u,(v2 - w1) - (v1 - w1) are_Ort_wrt w,y by A1, ANALMETR:10; (v2 - w1) - (v1 - w1) = v2 - (w1 + (v1 - w1)) by RLVECT_1:27 .= v2 - (v1 - (w1 - w1)) by RLVECT_1:29 .= v2 - (v1 - (0. V)) by RLVECT_1:15 .= v2 - v1 by RLVECT_1:13 ; hence u,v,v1,v2 are_Ort_wrt w,y by A3, ANALMETR:def_3; ::_thesis: verum end; Lm8: for V being RealLinearSpace for w, y, u, v being VECTOR of V st Gen w,y & u,v,u,v are_Ort_wrt w,y holds u = v proof let V be RealLinearSpace; ::_thesis: for w, y, u, v being VECTOR of V st Gen w,y & u,v,u,v are_Ort_wrt w,y holds u = v let w, y, u, v be VECTOR of V; ::_thesis: ( Gen w,y & u,v,u,v are_Ort_wrt w,y implies u = v ) assume that A1: Gen w,y and A2: u,v,u,v are_Ort_wrt w,y ; ::_thesis: u = v v - u,v - u are_Ort_wrt w,y by A2, ANALMETR:def_3; then v - u = 0. V by A1, ANALMETR:11; hence u = v by RLVECT_1:21; ::_thesis: verum end; Lm9: for V being RealLinearSpace for w, y, u, v, u1 being VECTOR of V st Gen w,y holds ( u,v,u1,u1 are_Ort_wrt w,y & u1,u1,u,v are_Ort_wrt w,y ) proof let V be RealLinearSpace; ::_thesis: for w, y, u, v, u1 being VECTOR of V st Gen w,y holds ( u,v,u1,u1 are_Ort_wrt w,y & u1,u1,u,v are_Ort_wrt w,y ) let w, y, u, v, u1 be VECTOR of V; ::_thesis: ( Gen w,y implies ( u,v,u1,u1 are_Ort_wrt w,y & u1,u1,u,v are_Ort_wrt w,y ) ) A1: u1 - u1 = 0. V by RLVECT_1:15; assume Gen w,y ; ::_thesis: ( u,v,u1,u1 are_Ort_wrt w,y & u1,u1,u,v are_Ort_wrt w,y ) then v - u,u1 - u1 are_Ort_wrt w,y by A1, ANALMETR:5; hence u,v,u1,u1 are_Ort_wrt w,y by ANALMETR:def_3; ::_thesis: u1,u1,u,v are_Ort_wrt w,y hence u1,u1,u,v are_Ort_wrt w,y by Lm5; ::_thesis: verum end; Lm10: for V being RealLinearSpace for w, y, u1, v1, u, v, u2, v2 being VECTOR of V st Gen w,y & ( u1,v1 '||' u,v or u,v '||' u1,v1 ) & ( u2,v2,u,v are_Ort_wrt w,y or u,v,u2,v2 are_Ort_wrt w,y ) & u <> v holds ( u1,v1,u2,v2 are_Ort_wrt w,y & u2,v2,u1,v1 are_Ort_wrt w,y ) proof let V be RealLinearSpace; ::_thesis: for w, y, u1, v1, u, v, u2, v2 being VECTOR of V st Gen w,y & ( u1,v1 '||' u,v or u,v '||' u1,v1 ) & ( u2,v2,u,v are_Ort_wrt w,y or u,v,u2,v2 are_Ort_wrt w,y ) & u <> v holds ( u1,v1,u2,v2 are_Ort_wrt w,y & u2,v2,u1,v1 are_Ort_wrt w,y ) let w, y, u1, v1, u, v, u2, v2 be VECTOR of V; ::_thesis: ( Gen w,y & ( u1,v1 '||' u,v or u,v '||' u1,v1 ) & ( u2,v2,u,v are_Ort_wrt w,y or u,v,u2,v2 are_Ort_wrt w,y ) & u <> v implies ( u1,v1,u2,v2 are_Ort_wrt w,y & u2,v2,u1,v1 are_Ort_wrt w,y ) ) assume that A1: Gen w,y and A2: ( u1,v1 '||' u,v or u,v '||' u1,v1 ) and A3: ( u2,v2,u,v are_Ort_wrt w,y or u,v,u2,v2 are_Ort_wrt w,y ) and A4: u <> v ; ::_thesis: ( u1,v1,u2,v2 are_Ort_wrt w,y & u2,v2,u1,v1 are_Ort_wrt w,y ) reconsider p9 = u, q9 = v, p19 = u1, q19 = v1, p29 = u2, q29 = v2 as Element of the carrier of (AMSpace (V,w,y)) by ANALMETR:19; reconsider S = AMSpace (V,w,y) as OrtAfSp by A1, ANALMETR:33; reconsider p = p9, q = q9, p1 = p19, q1 = q19, p2 = p29, q2 = q29 as Element of S ; A5: ( p2,q2 _|_ p,q or p,q _|_ p2,q2 ) by A3, ANALMETR:21; ( p1,q1 // p,q or p,q // p1,q1 ) by A2, Th4; then ( p1,q1 _|_ p2,q2 & p2,q2 _|_ p1,q1 ) by A4, A5, ANALMETR:62; hence ( u1,v1,u2,v2 are_Ort_wrt w,y & u2,v2,u1,v1 are_Ort_wrt w,y ) by ANALMETR:21; ::_thesis: verum end; definition let V be RealLinearSpace; let w, y, u, u1, v, v1 be VECTOR of V; predu,u1,v,v1 are_DTr_wrt w,y means :Def3: :: GEOMTRAP:def 3 ( u,u1 // v,v1 & u,u1,u # u1,v # v1 are_Ort_wrt w,y & v,v1,u # u1,v # v1 are_Ort_wrt w,y ); end; :: deftheorem Def3 defines are_DTr_wrt GEOMTRAP:def_3_:_ for V being RealLinearSpace for w, y, u, u1, v, v1 being VECTOR of V holds ( u,u1,v,v1 are_DTr_wrt w,y iff ( u,u1 // v,v1 & u,u1,u # u1,v # v1 are_Ort_wrt w,y & v,v1,u # u1,v # v1 are_Ort_wrt w,y ) ); theorem :: GEOMTRAP:15 for V being RealLinearSpace for w, y, u, v being VECTOR of V st Gen w,y holds u,u,v,v are_DTr_wrt w,y proof let V be RealLinearSpace; ::_thesis: for w, y, u, v being VECTOR of V st Gen w,y holds u,u,v,v are_DTr_wrt w,y let w, y, u, v be VECTOR of V; ::_thesis: ( Gen w,y implies u,u,v,v are_DTr_wrt w,y ) assume Gen w,y ; ::_thesis: u,u,v,v are_DTr_wrt w,y then A1: ( u,u,u # u,v # v are_Ort_wrt w,y & v,v,u # u,v # v are_Ort_wrt w,y ) by Lm5, Lm6; u,u // v,v by ANALOAF:9; hence u,u,v,v are_DTr_wrt w,y by A1, Def3; ::_thesis: verum end; theorem :: GEOMTRAP:16 for V being RealLinearSpace for w, y, u, v being VECTOR of V st Gen w,y holds u,v,u,v are_DTr_wrt w,y proof let V be RealLinearSpace; ::_thesis: for w, y, u, v being VECTOR of V st Gen w,y holds u,v,u,v are_DTr_wrt w,y let w, y, u, v be VECTOR of V; ::_thesis: ( Gen w,y implies u,v,u,v are_DTr_wrt w,y ) assume Gen w,y ; ::_thesis: u,v,u,v are_DTr_wrt w,y then ( u,v // u,v & u,v,u # v,u # v are_Ort_wrt w,y ) by Lm6, ANALOAF:8; hence u,v,u,v are_DTr_wrt w,y by Def3; ::_thesis: verum end; theorem Th17: :: GEOMTRAP:17 for V being RealLinearSpace for u, v, w, y being VECTOR of V st u,v,v,u are_DTr_wrt w,y holds u = v proof let V be RealLinearSpace; ::_thesis: for u, v, w, y being VECTOR of V st u,v,v,u are_DTr_wrt w,y holds u = v let u, v, w, y be VECTOR of V; ::_thesis: ( u,v,v,u are_DTr_wrt w,y implies u = v ) assume u,v,v,u are_DTr_wrt w,y ; ::_thesis: u = v then u,v // v,u by Def3; hence u = v by ANALOAF:10; ::_thesis: verum end; theorem Th18: :: GEOMTRAP:18 for V being RealLinearSpace for w, y, v1, u, v2 being VECTOR of V st Gen w,y & v1,u,u,v2 are_DTr_wrt w,y holds ( v1 = u & u = v2 ) proof let V be RealLinearSpace; ::_thesis: for w, y, v1, u, v2 being VECTOR of V st Gen w,y & v1,u,u,v2 are_DTr_wrt w,y holds ( v1 = u & u = v2 ) let w, y, v1, u, v2 be VECTOR of V; ::_thesis: ( Gen w,y & v1,u,u,v2 are_DTr_wrt w,y implies ( v1 = u & u = v2 ) ) assume that A1: Gen w,y and A2: v1,u,u,v2 are_DTr_wrt w,y ; ::_thesis: ( v1 = u & u = v2 ) set a = v1 # u; set b = u # v2; A3: v1,u,v1 # u,u # v2 are_Ort_wrt w,y by A2, Def3; A4: u,v2,v1 # u,u # v2 are_Ort_wrt w,y by A2, Def3; A5: v1,u // u,v2 by A2, Def3; percases ( v1 = v2 or v1 <> v2 ) ; suppose v1 = v2 ; ::_thesis: ( v1 = u & u = v2 ) hence ( v1 = u & u = v2 ) by A2, Th17; ::_thesis: verum end; suppose v1 <> v2 ; ::_thesis: ( v1 = u & u = v2 ) then A6: v1 # u <> u # v2 by Th7; u,v2 // u,u # v2 by Th12; then A7: u,v2 '||' u,u # v2 by Def1; A8: v1 # u,u // u,u # v2 by A5, Th13; then u,u # v2 // v1 # u,u # v2 by Lm1; then A9: u,u # v2 '||' v1 # u,u # v2 by Def1; A10: u = v2 proof assume A11: u <> v2 ; ::_thesis: contradiction A12: u <> u # v2 proof assume u = u # v2 ; ::_thesis: contradiction then u # v2 = u # u ; hence contradiction by A11, Th7; ::_thesis: verum end; u,u # v2,v1 # u,u # v2 are_Ort_wrt w,y by A1, A4, A7, A11, Lm10; then u,u # v2,u,u # v2 are_Ort_wrt w,y by A1, A6, A9, Lm10; hence contradiction by A1, A12, Lm8; ::_thesis: verum end; v1,u // v1 # u,u by Th12; then A13: v1,u '||' v1 # u,u by Def1; v1 # u,u // v1 # u,u # v2 by A8, Lm1; then A14: v1 # u,u '||' v1 # u,u # v2 by Def1; v1 = u proof assume A15: v1 <> u ; ::_thesis: contradiction A16: u <> v1 # u proof assume u = v1 # u ; ::_thesis: contradiction then v1 # u = u # u ; hence contradiction by A15, Th7; ::_thesis: verum end; v1 # u,u,v1 # u,u # v2 are_Ort_wrt w,y by A1, A3, A13, A15, Lm10; then v1 # u,u,v1 # u,u are_Ort_wrt w,y by A1, A6, A14, Lm10; hence contradiction by A1, A16, Lm8; ::_thesis: verum end; hence ( v1 = u & u = v2 ) by A10; ::_thesis: verum end; end; end; theorem Th19: :: GEOMTRAP:19 for V being RealLinearSpace for w, y, u, v, u1, v1, u2, v2 being VECTOR of V st Gen w,y & u,v,u1,v1 are_DTr_wrt w,y & u,v,u2,v2 are_DTr_wrt w,y & u <> v holds u1,v1,u2,v2 are_DTr_wrt w,y proof let V be RealLinearSpace; ::_thesis: for w, y, u, v, u1, v1, u2, v2 being VECTOR of V st Gen w,y & u,v,u1,v1 are_DTr_wrt w,y & u,v,u2,v2 are_DTr_wrt w,y & u <> v holds u1,v1,u2,v2 are_DTr_wrt w,y let w, y, u, v, u1, v1, u2, v2 be VECTOR of V; ::_thesis: ( Gen w,y & u,v,u1,v1 are_DTr_wrt w,y & u,v,u2,v2 are_DTr_wrt w,y & u <> v implies u1,v1,u2,v2 are_DTr_wrt w,y ) assume that A1: Gen w,y and A2: u,v,u1,v1 are_DTr_wrt w,y and A3: u,v,u2,v2 are_DTr_wrt w,y and A4: u <> v ; ::_thesis: u1,v1,u2,v2 are_DTr_wrt w,y set a = u1 # v1; set b = u2 # v2; ( u,v,u # v,u1 # v1 are_Ort_wrt w,y & u,v,u # v,u2 # v2 are_Ort_wrt w,y ) by A2, A3, Def3; then A5: u,v,u1 # v1,u2 # v2 are_Ort_wrt w,y by A1, Lm7; A6: u,v // u2,v2 by A3, Def3; then u,v '||' u2,v2 by Def1; then A7: u2,v2,u1 # v1,u2 # v2 are_Ort_wrt w,y by A1, A4, A5, Lm10; A8: u,v // u1,v1 by A2, Def3; then u,v '||' u1,v1 by Def1; then A9: u1,v1,u1 # v1,u2 # v2 are_Ort_wrt w,y by A1, A4, A5, Lm10; u1,v1 // u2,v2 by A4, A8, A6, ANALOAF:11; hence u1,v1,u2,v2 are_DTr_wrt w,y by A9, A7, Def3; ::_thesis: verum end; theorem Th20: :: GEOMTRAP:20 for V being RealLinearSpace for w, y, u, v, u1 being VECTOR of V st Gen w,y holds ex t being VECTOR of V st ( u,v,u1,t are_DTr_wrt w,y or u,v,t,u1 are_DTr_wrt w,y ) proof let V be RealLinearSpace; ::_thesis: for w, y, u, v, u1 being VECTOR of V st Gen w,y holds ex t being VECTOR of V st ( u,v,u1,t are_DTr_wrt w,y or u,v,t,u1 are_DTr_wrt w,y ) let w, y, u, v, u1 be VECTOR of V; ::_thesis: ( Gen w,y implies ex t being VECTOR of V st ( u,v,u1,t are_DTr_wrt w,y or u,v,t,u1 are_DTr_wrt w,y ) ) assume A1: Gen w,y ; ::_thesis: ex t being VECTOR of V st ( u,v,u1,t are_DTr_wrt w,y or u,v,t,u1 are_DTr_wrt w,y ) set a = u # v; percases ( u = v or u <> v ) ; supposeA2: u = v ; ::_thesis: ex t being VECTOR of V st ( u,v,u1,t are_DTr_wrt w,y or u,v,t,u1 are_DTr_wrt w,y ) A3: u1,u1,u # u,u1 # u1 are_Ort_wrt w,y by A1, Lm5, Lm6; ( u,u // u1,u1 & u,u,u # u,u1 # u1 are_Ort_wrt w,y ) by A1, Lm5, Lm6, ANALOAF:9; then u,u,u1,u1 are_DTr_wrt w,y by A3, Def3; hence ex t being VECTOR of V st ( u,v,u1,t are_DTr_wrt w,y or u,v,t,u1 are_DTr_wrt w,y ) by A2; ::_thesis: verum end; supposeA4: u <> v ; ::_thesis: ex t being VECTOR of V st ( u,v,u1,t are_DTr_wrt w,y or u,v,t,u1 are_DTr_wrt w,y ) u # v <> u proof assume u # v = u ; ::_thesis: contradiction then u # u = u # v ; hence contradiction by A4, Th7; ::_thesis: verum end; then u - (u # v) <> 0. V by RLVECT_1:21; then consider r being Real such that A5: (u1 - (u # v)) - (r * (u - (u # v))),u - (u # v) are_Ort_wrt w,y by A1, ANALMETR:13; set b = u1 - (r * (u - (u # v))); set t = (2 * (u1 - (r * (u - (u # v))))) - u1; u1 + ((2 * (u1 - (r * (u - (u # v))))) - u1) = (1 + 1) * (u1 - (r * (u - (u # v)))) by RLSUB_2:61 .= (1 * (u1 - (r * (u - (u # v))))) + (1 * (u1 - (r * (u - (u # v))))) by RLVECT_1:def_6 .= (u1 - (r * (u - (u # v)))) + (1 * (u1 - (r * (u - (u # v))))) by RLVECT_1:def_8 .= (u1 - (r * (u - (u # v)))) + (u1 - (r * (u - (u # v)))) by RLVECT_1:def_8 ; then A6: u1 # ((2 * (u1 - (r * (u - (u # v))))) - u1) = u1 - (r * (u - (u # v))) by Def2; u1 - (u1 - (r * (u - (u # v)))) = (u1 - u1) + (r * (u - (u # v))) by RLVECT_1:29 .= (0. V) + (r * (u - (u # v))) by RLVECT_1:15 .= r * (u - (u # v)) by RLVECT_1:4 ; then A7: u1 - ((2 * (u1 - (r * (u - (u # v))))) - u1) = 2 * (r * (u - (u # v))) by A6, Th10 .= (2 * r) * (u - (u # v)) by RLVECT_1:def_7 ; A8: u1 - ((u # v) + (r * (u - (u # v)))) = (u1 - (r * (u - (u # v)))) - (u # v) by RLVECT_1:27; then (u1 - (r * (u - (u # v)))) - (u # v),u - (u # v) are_Ort_wrt w,y by A5, RLVECT_1:27; then (u1 - (r * (u - (u # v)))) - (u # v),u1 - ((2 * (u1 - (r * (u - (u # v))))) - u1) are_Ort_wrt w,y by A7, ANALMETR:7; then A9: u # v,u1 - (r * (u - (u # v))),(2 * (u1 - (r * (u - (u # v))))) - u1,u1 are_Ort_wrt w,y by ANALMETR:def_3; then A10: (2 * (u1 - (r * (u - (u # v))))) - u1,u1,u # v,((2 * (u1 - (r * (u - (u # v))))) - u1) # u1 are_Ort_wrt w,y by A6, Lm5; A11: u - v = 2 * (u - (u # v)) by Th10; then u1 - ((2 * (u1 - (r * (u - (u # v))))) - u1) = r * (u - v) by A7, RLVECT_1:def_7; then r * (u - v) = 1 * (u1 - ((2 * (u1 - (r * (u - (u # v))))) - u1)) by RLVECT_1:def_8; then ( v,u // (2 * (u1 - (r * (u - (u # v))))) - u1,u1 or v,u // u1,(2 * (u1 - (r * (u - (u # v))))) - u1 ) by ANALMETR:14; then A12: ( u,v // u1,(2 * (u1 - (r * (u - (u # v))))) - u1 or u,v // (2 * (u1 - (r * (u - (u # v))))) - u1,u1 ) by ANALOAF:12; u # v,u1 - (r * (u - (u # v))),u1,(2 * (u1 - (r * (u - (u # v))))) - u1 are_Ort_wrt w,y by A9, Lm4; then A13: u1,(2 * (u1 - (r * (u - (u # v))))) - u1,u # v,u1 # ((2 * (u1 - (r * (u - (u # v))))) - u1) are_Ort_wrt w,y by A6, Lm5; (u1 - (r * (u - (u # v)))) - (u # v) = (u1 - (u # v)) - (r * (u - (u # v))) by A8, RLVECT_1:27; then (u1 - (r * (u - (u # v)))) - (u # v),u - v are_Ort_wrt w,y by A5, A11, ANALMETR:7; then u # v,u1 - (r * (u - (u # v))),v,u are_Ort_wrt w,y by ANALMETR:def_3; then u # v,u1 - (r * (u - (u # v))),u,v are_Ort_wrt w,y by Lm4; then u,v,u # v,u1 # ((2 * (u1 - (r * (u - (u # v))))) - u1) are_Ort_wrt w,y by A6, Lm5; then ( u,v,u1,(2 * (u1 - (r * (u - (u # v))))) - u1 are_DTr_wrt w,y or u,v,(2 * (u1 - (r * (u - (u # v))))) - u1,u1 are_DTr_wrt w,y ) by A13, A10, A12, Def3; hence ex t being VECTOR of V st ( u,v,u1,t are_DTr_wrt w,y or u,v,t,u1 are_DTr_wrt w,y ) ; ::_thesis: verum end; end; end; theorem Th21: :: GEOMTRAP:21 for V being RealLinearSpace for u, v, u1, v1, w, y being VECTOR of V st u,v,u1,v1 are_DTr_wrt w,y holds u1,v1,u,v are_DTr_wrt w,y proof let V be RealLinearSpace; ::_thesis: for u, v, u1, v1, w, y being VECTOR of V st u,v,u1,v1 are_DTr_wrt w,y holds u1,v1,u,v are_DTr_wrt w,y let u, v, u1, v1, w, y be VECTOR of V; ::_thesis: ( u,v,u1,v1 are_DTr_wrt w,y implies u1,v1,u,v are_DTr_wrt w,y ) assume A1: u,v,u1,v1 are_DTr_wrt w,y ; ::_thesis: u1,v1,u,v are_DTr_wrt w,y then u,v // u1,v1 by Def3; then A2: u1,v1 // u,v by ANALOAF:12; u1,v1,u # v,u1 # v1 are_Ort_wrt w,y by A1, Def3; then A3: u1,v1,u1 # v1,u # v are_Ort_wrt w,y by Lm4; u,v,u # v,u1 # v1 are_Ort_wrt w,y by A1, Def3; then u,v,u1 # v1,u # v are_Ort_wrt w,y by Lm4; hence u1,v1,u,v are_DTr_wrt w,y by A2, A3, Def3; ::_thesis: verum end; theorem Th22: :: GEOMTRAP:22 for V being RealLinearSpace for u, v, u1, v1, w, y being VECTOR of V st u,v,u1,v1 are_DTr_wrt w,y holds v,u,v1,u1 are_DTr_wrt w,y proof let V be RealLinearSpace; ::_thesis: for u, v, u1, v1, w, y being VECTOR of V st u,v,u1,v1 are_DTr_wrt w,y holds v,u,v1,u1 are_DTr_wrt w,y let u, v, u1, v1, w, y be VECTOR of V; ::_thesis: ( u,v,u1,v1 are_DTr_wrt w,y implies v,u,v1,u1 are_DTr_wrt w,y ) assume A1: u,v,u1,v1 are_DTr_wrt w,y ; ::_thesis: v,u,v1,u1 are_DTr_wrt w,y then u,v // u1,v1 by Def3; then A2: v,u // v1,u1 by ANALOAF:12; A3: now__::_thesis:_for_u,_u9,_v,_v9_being_VECTOR_of_V_st_u,u9,v,v9_are_Ort_wrt_w,y_holds_ u9,u,v,v9_are_Ort_wrt_w,y let u, u9, v, v9 be VECTOR of V; ::_thesis: ( u,u9,v,v9 are_Ort_wrt w,y implies u9,u,v,v9 are_Ort_wrt w,y ) assume u,u9,v,v9 are_Ort_wrt w,y ; ::_thesis: u9,u,v,v9 are_Ort_wrt w,y then v,v9,u,u9 are_Ort_wrt w,y by Lm5; then v,v9,u9,u are_Ort_wrt w,y by Lm4; hence u9,u,v,v9 are_Ort_wrt w,y by Lm5; ::_thesis: verum end; u1,v1,u # v,u1 # v1 are_Ort_wrt w,y by A1, Def3; then A4: v1,u1,v # u,v1 # u1 are_Ort_wrt w,y by A3; u,v,u # v,u1 # v1 are_Ort_wrt w,y by A1, Def3; then v,u,v # u,v1 # u1 are_Ort_wrt w,y by A3; hence v,u,v1,u1 are_DTr_wrt w,y by A2, A4, Def3; ::_thesis: verum end; Lm11: for V being RealLinearSpace for w, y, u, v, u1, v1, u2, v2 being VECTOR of V st Gen w,y & u <> v & u,v '||' u,u1 & u,v '||' u,v1 & u,v '||' u,u2 & u,v '||' u,v2 holds u1,v1 '||' u2,v2 proof let V be RealLinearSpace; ::_thesis: for w, y, u, v, u1, v1, u2, v2 being VECTOR of V st Gen w,y & u <> v & u,v '||' u,u1 & u,v '||' u,v1 & u,v '||' u,u2 & u,v '||' u,v2 holds u1,v1 '||' u2,v2 let w, y, u, v, u1, v1, u2, v2 be VECTOR of V; ::_thesis: ( Gen w,y & u <> v & u,v '||' u,u1 & u,v '||' u,v1 & u,v '||' u,u2 & u,v '||' u,v2 implies u1,v1 '||' u2,v2 ) assume that A1: Gen w,y and A2: u <> v and A3: u,v '||' u,u1 and A4: u,v '||' u,v1 and A5: u,v '||' u,u2 and A6: u,v '||' u,v2 ; ::_thesis: u1,v1 '||' u2,v2 reconsider p9 = u, q9 = v, p19 = u1, q19 = v1, p29 = u2, q29 = v2 as Element of (Lambda (OASpace V)) by ANALMETR:16; reconsider S9 = OASpace V as OAffinSpace by A1, Th1; reconsider S = Lambda S9 as AffinSpace by DIRAF:41; reconsider p = p9, q = q9, p1 = p19, q1 = q19, p2 = p29, q2 = q29 as Element of the carrier of S ; p,q // p,p1 by A1, A3, Th3; then A7: LIN p,q,p1 by AFF_1:def_1; p,q // p,q1 by A1, A4, Th3; then A8: LIN p,q,q1 by AFF_1:def_1; p,q // p,q2 by A1, A6, Th3; then LIN p,q,q2 by AFF_1:def_1; then A9: LIN p1,q1,q2 by A2, A7, A8, AFF_1:8; p,q // p,p2 by A1, A5, Th3; then LIN p,q,p2 by AFF_1:def_1; then LIN p1,q1,p2 by A2, A7, A8, AFF_1:8; then p1,q1 // p2,q2 by A9, AFF_1:10; hence u1,v1 '||' u2,v2 by A1, Th3; ::_thesis: verum end; theorem Th23: :: GEOMTRAP:23 for V being RealLinearSpace for w, y, v, u1, u2 being VECTOR of V st Gen w,y & v,u1,v,u2 are_DTr_wrt w,y holds u1 = u2 proof let V be RealLinearSpace; ::_thesis: for w, y, v, u1, u2 being VECTOR of V st Gen w,y & v,u1,v,u2 are_DTr_wrt w,y holds u1 = u2 let w, y, v, u1, u2 be VECTOR of V; ::_thesis: ( Gen w,y & v,u1,v,u2 are_DTr_wrt w,y implies u1 = u2 ) assume that A1: Gen w,y and A2: v,u1,v,u2 are_DTr_wrt w,y ; ::_thesis: u1 = u2 A3: v,u1,v # u1,v # u2 are_Ort_wrt w,y by A2, Def3; A4: v,u2,v # u1,v # u2 are_Ort_wrt w,y by A2, Def3; v,u1 // v,u1 by ANALOAF:8; then A5: v,u1 '||' v,u1 by Def1; set b = v # u1; set c = v # u2; A6: v,u1 // v,v # u1 by Th12; then A7: v,u1 '||' v,v # u1 by Def1; A8: v,u1 // v,u2 by A2, Def3; v,u2 // v,v # u1 proof percases ( v = v # u1 or v <> v # u1 ) ; suppose v = v # u1 ; ::_thesis: v,u2 // v,v # u1 hence v,u2 // v,v # u1 by ANALOAF:9; ::_thesis: verum end; suppose v <> v # u1 ; ::_thesis: v,u2 // v,v # u1 then v <> u1 ; hence v,u2 // v,v # u1 by A8, A6, ANALOAF:11; ::_thesis: verum end; end; end; then A9: v,u2 '||' v,v # u1 by Def1; v,u2 // v,v by ANALOAF:9; then A10: v,u2 '||' v,v by Def1; A11: v,u2 // v,v # u2 by Th12; then A12: v,u2 '||' v,v # u2 by Def1; A13: v,u2 // v,u1 by A8, ANALOAF:12; v,u1 // v,v # u2 proof percases ( v = v # u2 or v <> v # u2 ) ; suppose v = v # u2 ; ::_thesis: v,u1 // v,v # u2 hence v,u1 // v,v # u2 by ANALOAF:9; ::_thesis: verum end; suppose v <> v # u2 ; ::_thesis: v,u1 // v,v # u2 then v <> u2 ; hence v,u1 // v,v # u2 by A13, A11, ANALOAF:11; ::_thesis: verum end; end; end; then A14: v,u1 '||' v,v # u2 by Def1; v,u2 // v,u2 by ANALOAF:8; then A15: v,u2 '||' v,u2 by Def1; v,u1 // v,v by ANALOAF:9; then A16: v,u1 '||' v,v by Def1; assume A17: u1 <> u2 ; ::_thesis: contradiction percases ( v <> u1 or v <> u2 ) by A17; supposeA18: v <> u1 ; ::_thesis: contradiction then v,u1 '||' v # u1,v # u2 by A1, A7, A5, A16, A14, Lm11; then v # u1,v # u2,v # u1,v # u2 are_Ort_wrt w,y by A1, A3, A18, Lm10; hence contradiction by A1, A17, Lm8, Th7; ::_thesis: verum end; supposeA19: v <> u2 ; ::_thesis: contradiction then v,u2 '||' v # u1,v # u2 by A1, A12, A15, A10, A9, Lm11; then v # u1,v # u2,v # u1,v # u2 are_Ort_wrt w,y by A1, A4, A19, Lm10; hence contradiction by A1, A17, Lm8, Th7; ::_thesis: verum end; end; end; theorem Th24: :: GEOMTRAP:24 for V being RealLinearSpace for w, y, u, v, u1, v1, v2 being VECTOR of V st Gen w,y & u,v,u1,v1 are_DTr_wrt w,y & u,v,u1,v2 are_DTr_wrt w,y & not u = v holds v1 = v2 proof let V be RealLinearSpace; ::_thesis: for w, y, u, v, u1, v1, v2 being VECTOR of V st Gen w,y & u,v,u1,v1 are_DTr_wrt w,y & u,v,u1,v2 are_DTr_wrt w,y & not u = v holds v1 = v2 let w, y, u, v, u1, v1, v2 be VECTOR of V; ::_thesis: ( Gen w,y & u,v,u1,v1 are_DTr_wrt w,y & u,v,u1,v2 are_DTr_wrt w,y & not u = v implies v1 = v2 ) assume that A1: Gen w,y and A2: ( u,v,u1,v1 are_DTr_wrt w,y & u,v,u1,v2 are_DTr_wrt w,y ) ; ::_thesis: ( u = v or v1 = v2 ) assume u <> v ; ::_thesis: v1 = v2 then u1,v1,u1,v2 are_DTr_wrt w,y by A1, A2, Th19; hence v1 = v2 by A1, Th23; ::_thesis: verum end; theorem Th25: :: GEOMTRAP:25 for V being RealLinearSpace for w, y, u, u1, v, v1, v2 being VECTOR of V st Gen w,y & u <> u1 & u,u1,v,v1 are_DTr_wrt w,y & ( u,u1,v,v2 are_DTr_wrt w,y or u,u1,v2,v are_DTr_wrt w,y ) holds v1 = v2 proof let V be RealLinearSpace; ::_thesis: for w, y, u, u1, v, v1, v2 being VECTOR of V st Gen w,y & u <> u1 & u,u1,v,v1 are_DTr_wrt w,y & ( u,u1,v,v2 are_DTr_wrt w,y or u,u1,v2,v are_DTr_wrt w,y ) holds v1 = v2 let w, y, u, u1, v, v1, v2 be VECTOR of V; ::_thesis: ( Gen w,y & u <> u1 & u,u1,v,v1 are_DTr_wrt w,y & ( u,u1,v,v2 are_DTr_wrt w,y or u,u1,v2,v are_DTr_wrt w,y ) implies v1 = v2 ) assume that A1: Gen w,y and A2: ( u <> u1 & u,u1,v,v1 are_DTr_wrt w,y ) and A3: ( u,u1,v,v2 are_DTr_wrt w,y or u,u1,v2,v are_DTr_wrt w,y ) ; ::_thesis: v1 = v2 now__::_thesis:_(_u,u1,v2,v_are_DTr_wrt_w,y_implies_v1_=_v2_) assume u,u1,v2,v are_DTr_wrt w,y ; ::_thesis: v1 = v2 then A4: v2,v,v,v1 are_DTr_wrt w,y by A1, A2, Th19; then v = v2 by A1, Th18; hence v1 = v2 by A1, A4, Th18; ::_thesis: verum end; hence v1 = v2 by A1, A2, A3, Th24; ::_thesis: verum end; theorem Th26: :: GEOMTRAP:26 for V being RealLinearSpace for w, y, u, v, u1, v1 being VECTOR of V st Gen w,y & u,v,u1,v1 are_DTr_wrt w,y holds u,v,u # u1,v # v1 are_DTr_wrt w,y proof let V be RealLinearSpace; ::_thesis: for w, y, u, v, u1, v1 being VECTOR of V st Gen w,y & u,v,u1,v1 are_DTr_wrt w,y holds u,v,u # u1,v # v1 are_DTr_wrt w,y let w, y, u, v, u1, v1 be VECTOR of V; ::_thesis: ( Gen w,y & u,v,u1,v1 are_DTr_wrt w,y implies u,v,u # u1,v # v1 are_DTr_wrt w,y ) assume that A1: Gen w,y and A2: u,v,u1,v1 are_DTr_wrt w,y ; ::_thesis: u,v,u # u1,v # v1 are_DTr_wrt w,y set p = u # u1; set q = v # v1; set r = u # v; set s = u1 # v1; A3: ( u = v & u1 = v1 implies u # u1,v # v1,u # v,(u # u1) # (v # v1) are_Ort_wrt w,y ) by A1, Lm9; (u # u1) # (v # v1) = (u # v) # (u1 # v1) by Th6; then u # v,u1 # v1 // u # v,(u # u1) # (v # v1) by Th12; then A4: u # v,u1 # v1 '||' u # v,(u # u1) # (v # v1) by Def1; ( u,v,u # v,u1 # v1 are_Ort_wrt w,y & u1,v1,u # v,u1 # v1 are_Ort_wrt w,y ) by A2, Def3; then A5: ( u # v <> u1 # v1 implies ( u1,v1,u # v,(u # u1) # (v # v1) are_Ort_wrt w,y & u,v,u # v,(u # u1) # (v # v1) are_Ort_wrt w,y ) ) by A1, A4, Lm10; A6: now__::_thesis:_(_u_#_v_=_u1_#_v1_implies_(_u1,v1,u_#_v,(u_#_u1)_#_(v_#_v1)_are_Ort_wrt_w,y_&_u,v,u_#_v,(u_#_u1)_#_(v_#_v1)_are_Ort_wrt_w,y_)_) assume u # v = u1 # v1 ; ::_thesis: ( u1,v1,u # v,(u # u1) # (v # v1) are_Ort_wrt w,y & u,v,u # v,(u # u1) # (v # v1) are_Ort_wrt w,y ) then u # v = (u # v) # (u1 # v1) .= (u # u1) # (v # v1) by Th6 ; hence ( u1,v1,u # v,(u # u1) # (v # v1) are_Ort_wrt w,y & u,v,u # v,(u # u1) # (v # v1) are_Ort_wrt w,y ) by A1, Lm9; ::_thesis: verum end; A7: u,v // u1,v1 by A2, Def3; then u1,v1 // u,v by ANALOAF:12; then u1,v1 // u1 # u,v1 # v by Th14; then u1,v1 '||' u # u1,v # v1 by Def1; then A8: ( u = v & u1 <> v1 implies u # u1,v # v1,u # v,(u # u1) # (v # v1) are_Ort_wrt w,y ) by A1, A6, A5, Lm10; A9: u,v // u # u1,v # v1 by A7, Th14; then u,v '||' u # u1,v # v1 by Def1; then ( u <> v implies u # u1,v # v1,u # v,(u # u1) # (v # v1) are_Ort_wrt w,y ) by A1, A6, A5, Lm10; hence u,v,u # u1,v # v1 are_DTr_wrt w,y by A9, A6, A5, A8, A3, Def3; ::_thesis: verum end; theorem Th27: :: GEOMTRAP:27 for V being RealLinearSpace for w, y, u, v, u1, v1 being VECTOR of V st Gen w,y & u,v,u1,v1 are_DTr_wrt w,y & not u,v,u # v1,v # u1 are_DTr_wrt w,y holds u,v,v # u1,u # v1 are_DTr_wrt w,y proof let V be RealLinearSpace; ::_thesis: for w, y, u, v, u1, v1 being VECTOR of V st Gen w,y & u,v,u1,v1 are_DTr_wrt w,y & not u,v,u # v1,v # u1 are_DTr_wrt w,y holds u,v,v # u1,u # v1 are_DTr_wrt w,y let w, y, u, v, u1, v1 be VECTOR of V; ::_thesis: ( Gen w,y & u,v,u1,v1 are_DTr_wrt w,y & not u,v,u # v1,v # u1 are_DTr_wrt w,y implies u,v,v # u1,u # v1 are_DTr_wrt w,y ) assume that A1: Gen w,y and A2: u,v,u1,v1 are_DTr_wrt w,y ; ::_thesis: ( u,v,u # v1,v # u1 are_DTr_wrt w,y or u,v,v # u1,u # v1 are_DTr_wrt w,y ) set p = u # v1; set q = v # u1; set r = u # v; set s = u1 # v1; A3: u,v,u # v,u1 # v1 are_Ort_wrt w,y by A2, Def3; A4: (u # v1) # (v # u1) = (u # v) # (u1 # v1) by Th6; then u # v,u1 # v1 // u # v,(u # v1) # (v # u1) by Th12; then A5: u # v,u1 # v1 '||' u # v,(u # v1) # (v # u1) by Def1; u,v // u1,v1 by A2, Def3; then A6: u,v '||' u # v1,v # u1 by Lm2; then A7: ( u,v // u # v1,v # u1 or u,v // v # u1,u # v1 ) by Def1; percases ( u = v or u <> v ) ; suppose u = v ; ::_thesis: ( u,v,u # v1,v # u1 are_DTr_wrt w,y or u,v,v # u1,u # v1 are_DTr_wrt w,y ) hence ( u,v,u # v1,v # u1 are_DTr_wrt w,y or u,v,v # u1,u # v1 are_DTr_wrt w,y ) by A1, A2, Th26; ::_thesis: verum end; supposeA8: u <> v ; ::_thesis: ( u,v,u # v1,v # u1 are_DTr_wrt w,y or u,v,v # u1,u # v1 are_DTr_wrt w,y ) percases ( u # v = u1 # v1 or u # v <> u1 # v1 ) ; supposeA9: u # v = u1 # v1 ; ::_thesis: ( u,v,u # v1,v # u1 are_DTr_wrt w,y or u,v,v # u1,u # v1 are_DTr_wrt w,y ) then A10: v # u1,u # v1,u # v,(v # u1) # (u # v1) are_Ort_wrt w,y by A1, A4, Lm9; ( u,v,u # v,(u # v1) # (v # u1) are_Ort_wrt w,y & u # v1,v # u1,u # v,(u # v1) # (v # u1) are_Ort_wrt w,y ) by A1, A4, A9, Lm9; hence ( u,v,u # v1,v # u1 are_DTr_wrt w,y or u,v,v # u1,u # v1 are_DTr_wrt w,y ) by A7, A10, Def3; ::_thesis: verum end; suppose u # v <> u1 # v1 ; ::_thesis: ( u,v,u # v1,v # u1 are_DTr_wrt w,y or u,v,v # u1,u # v1 are_DTr_wrt w,y ) then A11: u,v,u # v,(u # v1) # (v # u1) are_Ort_wrt w,y by A1, A3, A5, Lm10; then u # v,(u # v1) # (v # u1),u # v1,v # u1 are_Ort_wrt w,y by A1, A6, A8, Lm10; then u # v,(u # v1) # (v # u1),v # u1,u # v1 are_Ort_wrt w,y by Lm4; then A12: v # u1,u # v1,u # v,(v # u1) # (u # v1) are_Ort_wrt w,y by Lm5; u # v1,v # u1,u # v,(u # v1) # (v # u1) are_Ort_wrt w,y by A1, A6, A8, A11, Lm10; hence ( u,v,u # v1,v # u1 are_DTr_wrt w,y or u,v,v # u1,u # v1 are_DTr_wrt w,y ) by A7, A11, A12, Def3; ::_thesis: verum end; end; end; end; end; theorem Th28: :: GEOMTRAP:28 for V being RealLinearSpace for w, y, u, u1, u2, v1, v2, t1, t2, w1, w2 being VECTOR of V st u = u1 # t1 & u = u2 # t2 & u = v1 # w1 & u = v2 # w2 & u1,u2,v1,v2 are_DTr_wrt w,y holds t1,t2,w1,w2 are_DTr_wrt w,y proof let V be RealLinearSpace; ::_thesis: for w, y, u, u1, u2, v1, v2, t1, t2, w1, w2 being VECTOR of V st u = u1 # t1 & u = u2 # t2 & u = v1 # w1 & u = v2 # w2 & u1,u2,v1,v2 are_DTr_wrt w,y holds t1,t2,w1,w2 are_DTr_wrt w,y let w, y be VECTOR of V; ::_thesis: for u, u1, u2, v1, v2, t1, t2, w1, w2 being VECTOR of V st u = u1 # t1 & u = u2 # t2 & u = v1 # w1 & u = v2 # w2 & u1,u2,v1,v2 are_DTr_wrt w,y holds t1,t2,w1,w2 are_DTr_wrt w,y let u, u1, u2, v1, v2, t1, t2, w1, w2 be VECTOR of V; ::_thesis: ( u = u1 # t1 & u = u2 # t2 & u = v1 # w1 & u = v2 # w2 & u1,u2,v1,v2 are_DTr_wrt w,y implies t1,t2,w1,w2 are_DTr_wrt w,y ) assume that A1: ( u = u1 # t1 & u = u2 # t2 ) and A2: ( u = v1 # w1 & u = v2 # w2 ) and A3: u1,u2,v1,v2 are_DTr_wrt w,y ; ::_thesis: t1,t2,w1,w2 are_DTr_wrt w,y A4: u1,u2 // v1,v2 by A3, Def3; set p = u1 # u2; set q = v1 # v2; set r = t1 # t2; set s = w1 # w2; A5: (v1 # v2) # (w1 # w2) = u # u by A2, Th6 .= u ; (u1 # u2) # (t1 # t2) = u # u by A1, Th6 .= u ; then A6: (w1 # w2) - (t1 # t2) = - ((v1 # v2) - (u1 # u2)) by A5, Lm3 .= (- 1) * ((v1 # v2) - (u1 # u2)) by RLVECT_1:16 ; A7: ( u2 - u1 = - (t2 - t1) & v2 - v1 = - (w2 - w1) ) by A1, A2, Lm3; A8: t1,t2 // w1,w2 proof percases ( u1 = u2 or v1 = v2 or ( u1 <> u2 & v1 <> v2 ) ) ; suppose u1 = u2 ; ::_thesis: t1,t2 // w1,w2 then t1 = t2 by A1, Th7; hence t1,t2 // w1,w2 by ANALOAF:9; ::_thesis: verum end; suppose v1 = v2 ; ::_thesis: t1,t2 // w1,w2 then w1 = w2 by A2, Th7; hence t1,t2 // w1,w2 by ANALOAF:9; ::_thesis: verum end; suppose ( u1 <> u2 & v1 <> v2 ) ; ::_thesis: t1,t2 // w1,w2 then consider a, b being Real such that A9: ( 0 < a & 0 < b ) and A10: a * (u2 - u1) = b * (v2 - v1) by A4, ANALOAF:def_1; a * (t2 - t1) = a * (- (- (t2 - t1))) by RLVECT_1:17 .= - (b * (- (w2 - w1))) by A7, A10, RLVECT_1:25 .= b * (- (- (w2 - w1))) by RLVECT_1:25 .= b * (w2 - w1) by RLVECT_1:17 ; hence t1,t2 // w1,w2 by A9, ANALOAF:def_1; ::_thesis: verum end; end; end; w2 - w1 = - (v2 - v1) by A2, Lm3; then A11: w2 - w1 = (- 1) * (v2 - v1) by RLVECT_1:16; v1,v2,u1 # u2,v1 # v2 are_Ort_wrt w,y by A3, Def3; then v2 - v1,(v1 # v2) - (u1 # u2) are_Ort_wrt w,y by ANALMETR:def_3; then w2 - w1,(w1 # w2) - (t1 # t2) are_Ort_wrt w,y by A6, A11, ANALMETR:6; then A12: w1,w2,t1 # t2,w1 # w2 are_Ort_wrt w,y by ANALMETR:def_3; t2 - t1 = - (u2 - u1) by A1, Lm3; then A13: t2 - t1 = (- 1) * (u2 - u1) by RLVECT_1:16; u1,u2,u1 # u2,v1 # v2 are_Ort_wrt w,y by A3, Def3; then u2 - u1,(v1 # v2) - (u1 # u2) are_Ort_wrt w,y by ANALMETR:def_3; then t2 - t1,(w1 # w2) - (t1 # t2) are_Ort_wrt w,y by A6, A13, ANALMETR:6; then t1,t2,t1 # t2,w1 # w2 are_Ort_wrt w,y by ANALMETR:def_3; hence t1,t2,w1,w2 are_DTr_wrt w,y by A8, A12, Def3; ::_thesis: verum end; Lm12: for V being RealLinearSpace for v1, w, y, v2 being VECTOR of V for b1, b2, c1, c2 being Real st v1 = (b1 * w) + (b2 * y) & v2 = (c1 * w) + (c2 * y) holds ( v1 + v2 = ((b1 + c1) * w) + ((b2 + c2) * y) & v1 - v2 = ((b1 - c1) * w) + ((b2 - c2) * y) ) proof let V be RealLinearSpace; ::_thesis: for v1, w, y, v2 being VECTOR of V for b1, b2, c1, c2 being Real st v1 = (b1 * w) + (b2 * y) & v2 = (c1 * w) + (c2 * y) holds ( v1 + v2 = ((b1 + c1) * w) + ((b2 + c2) * y) & v1 - v2 = ((b1 - c1) * w) + ((b2 - c2) * y) ) let v1, w, y, v2 be VECTOR of V; ::_thesis: for b1, b2, c1, c2 being Real st v1 = (b1 * w) + (b2 * y) & v2 = (c1 * w) + (c2 * y) holds ( v1 + v2 = ((b1 + c1) * w) + ((b2 + c2) * y) & v1 - v2 = ((b1 - c1) * w) + ((b2 - c2) * y) ) let b1, b2, c1, c2 be Real; ::_thesis: ( v1 = (b1 * w) + (b2 * y) & v2 = (c1 * w) + (c2 * y) implies ( v1 + v2 = ((b1 + c1) * w) + ((b2 + c2) * y) & v1 - v2 = ((b1 - c1) * w) + ((b2 - c2) * y) ) ) assume A1: ( v1 = (b1 * w) + (b2 * y) & v2 = (c1 * w) + (c2 * y) ) ; ::_thesis: ( v1 + v2 = ((b1 + c1) * w) + ((b2 + c2) * y) & v1 - v2 = ((b1 - c1) * w) + ((b2 - c2) * y) ) hence v1 + v2 = (((b1 * w) + (b2 * y)) + (c1 * w)) + (c2 * y) by RLVECT_1:def_3 .= (((b1 * w) + (c1 * w)) + (b2 * y)) + (c2 * y) by RLVECT_1:def_3 .= (((b1 + c1) * w) + (b2 * y)) + (c2 * y) by RLVECT_1:def_6 .= ((b1 + c1) * w) + ((b2 * y) + (c2 * y)) by RLVECT_1:def_3 .= ((b1 + c1) * w) + ((b2 + c2) * y) by RLVECT_1:def_6 ; ::_thesis: v1 - v2 = ((b1 - c1) * w) + ((b2 - c2) * y) thus v1 - v2 = ((b1 * w) + (b2 * y)) + ((- (c1 * w)) + (- (c2 * y))) by A1, RLVECT_1:31 .= ((b1 * w) + (b2 * y)) + ((c1 * (- w)) + (- (c2 * y))) by RLVECT_1:25 .= ((b1 * w) + (b2 * y)) + ((c1 * (- w)) + (c2 * (- y))) by RLVECT_1:25 .= ((b1 * w) + (b2 * y)) + (((- c1) * w) + (c2 * (- y))) by RLVECT_1:24 .= ((b1 * w) + (b2 * y)) + (((- c1) * w) + ((- c2) * y)) by RLVECT_1:24 .= (((b1 * w) + (b2 * y)) + ((- c1) * w)) + ((- c2) * y) by RLVECT_1:def_3 .= (((b1 * w) + ((- c1) * w)) + (b2 * y)) + ((- c2) * y) by RLVECT_1:def_3 .= (((b1 + (- c1)) * w) + (b2 * y)) + ((- c2) * y) by RLVECT_1:def_6 .= ((b1 + (- c1)) * w) + ((b2 * y) + ((- c2) * y)) by RLVECT_1:def_3 .= ((b1 - c1) * w) + ((b2 + (- c2)) * y) by RLVECT_1:def_6 .= ((b1 - c1) * w) + ((b2 - c2) * y) ; ::_thesis: verum end; Lm13: for V being RealLinearSpace for v, w, y being VECTOR of V for b1, b2, a being Real st v = (b1 * w) + (b2 * y) holds a * v = ((a * b1) * w) + ((a * b2) * y) proof let V be RealLinearSpace; ::_thesis: for v, w, y being VECTOR of V for b1, b2, a being Real st v = (b1 * w) + (b2 * y) holds a * v = ((a * b1) * w) + ((a * b2) * y) let v, w, y be VECTOR of V; ::_thesis: for b1, b2, a being Real st v = (b1 * w) + (b2 * y) holds a * v = ((a * b1) * w) + ((a * b2) * y) let b1, b2, a be Real; ::_thesis: ( v = (b1 * w) + (b2 * y) implies a * v = ((a * b1) * w) + ((a * b2) * y) ) assume v = (b1 * w) + (b2 * y) ; ::_thesis: a * v = ((a * b1) * w) + ((a * b2) * y) hence a * v = (a * (b1 * w)) + (a * (b2 * y)) by RLVECT_1:def_5 .= ((a * b1) * w) + (a * (b2 * y)) by RLVECT_1:def_7 .= ((a * b1) * w) + ((a * b2) * y) by RLVECT_1:def_7 ; ::_thesis: verum end; Lm14: for V being RealLinearSpace for w, y being VECTOR of V for a1, a2, b1, b2 being Real st Gen w,y & (a1 * w) + (a2 * y) = (b1 * w) + (b2 * y) holds ( a1 = b1 & a2 = b2 ) proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V for a1, a2, b1, b2 being Real st Gen w,y & (a1 * w) + (a2 * y) = (b1 * w) + (b2 * y) holds ( a1 = b1 & a2 = b2 ) let w, y be VECTOR of V; ::_thesis: for a1, a2, b1, b2 being Real st Gen w,y & (a1 * w) + (a2 * y) = (b1 * w) + (b2 * y) holds ( a1 = b1 & a2 = b2 ) let a1, a2, b1, b2 be Real; ::_thesis: ( Gen w,y & (a1 * w) + (a2 * y) = (b1 * w) + (b2 * y) implies ( a1 = b1 & a2 = b2 ) ) assume that A1: Gen w,y and A2: (a1 * w) + (a2 * y) = (b1 * w) + (b2 * y) ; ::_thesis: ( a1 = b1 & a2 = b2 ) 0. V = ((a1 * w) + (a2 * y)) - ((b1 * w) + (b2 * y)) by A2, RLVECT_1:15 .= ((a1 - b1) * w) + ((a2 - b2) * y) by Lm12 ; then ( (- b1) + a1 = 0 & (- b2) + a2 = 0 ) by A1, ANALMETR:def_1; hence ( a1 = b1 & a2 = b2 ) ; ::_thesis: verum end; definition let V be RealLinearSpace; let w, y, u be VECTOR of V; assume A1: Gen w,y ; func pr1 (w,y,u) -> Real means :Def4: :: GEOMTRAP:def 4 ex b being Real st u = (it * w) + (b * y); existence ex b1, b being Real st u = (b1 * w) + (b * y) by A1, ANALMETR:def_1; uniqueness for b1, b2 being Real st ex b being Real st u = (b1 * w) + (b * y) & ex b being Real st u = (b2 * w) + (b * y) holds b1 = b2 by A1, Lm14; end; :: deftheorem Def4 defines pr1 GEOMTRAP:def_4_:_ for V being RealLinearSpace for w, y, u being VECTOR of V st Gen w,y holds for b5 being Real holds ( b5 = pr1 (w,y,u) iff ex b being Real st u = (b5 * w) + (b * y) ); definition let V be RealLinearSpace; let w, y, u be VECTOR of V; assume B1: Gen w,y ; func pr2 (w,y,u) -> Real means :Def5: :: GEOMTRAP:def 5 ex a being Real st u = (a * w) + (it * y); existence ex b1, a being Real st u = (a * w) + (b1 * y) proof consider a, b being Real such that A1: u = (a * w) + (b * y) by B1, ANALMETR:def_1; take b ; ::_thesis: ex a being Real st u = (a * w) + (b * y) thus ex a being Real st u = (a * w) + (b * y) by A1; ::_thesis: verum end; uniqueness for b1, b2 being Real st ex a being Real st u = (a * w) + (b1 * y) & ex a being Real st u = (a * w) + (b2 * y) holds b1 = b2 by B1, Lm14; end; :: deftheorem Def5 defines pr2 GEOMTRAP:def_5_:_ for V being RealLinearSpace for w, y, u being VECTOR of V st Gen w,y holds for b5 being Real holds ( b5 = pr2 (w,y,u) iff ex a being Real st u = (a * w) + (b5 * y) ); Lm15: for V being RealLinearSpace for w, y, u being VECTOR of V st Gen w,y holds u = ((pr1 (w,y,u)) * w) + ((pr2 (w,y,u)) * y) proof let V be RealLinearSpace; ::_thesis: for w, y, u being VECTOR of V st Gen w,y holds u = ((pr1 (w,y,u)) * w) + ((pr2 (w,y,u)) * y) let w, y, u be VECTOR of V; ::_thesis: ( Gen w,y implies u = ((pr1 (w,y,u)) * w) + ((pr2 (w,y,u)) * y) ) assume A1: Gen w,y ; ::_thesis: u = ((pr1 (w,y,u)) * w) + ((pr2 (w,y,u)) * y) then ex b being Real st u = ((pr1 (w,y,u)) * w) + (b * y) by Def4; hence u = ((pr1 (w,y,u)) * w) + ((pr2 (w,y,u)) * y) by A1, Def5; ::_thesis: verum end; Lm16: for V being RealLinearSpace for w, y, u being VECTOR of V for a, b being Real st Gen w,y & u = (a * w) + (b * y) holds ( a = pr1 (w,y,u) & b = pr2 (w,y,u) ) proof let V be RealLinearSpace; ::_thesis: for w, y, u being VECTOR of V for a, b being Real st Gen w,y & u = (a * w) + (b * y) holds ( a = pr1 (w,y,u) & b = pr2 (w,y,u) ) let w, y, u be VECTOR of V; ::_thesis: for a, b being Real st Gen w,y & u = (a * w) + (b * y) holds ( a = pr1 (w,y,u) & b = pr2 (w,y,u) ) let a, b be Real; ::_thesis: ( Gen w,y & u = (a * w) + (b * y) implies ( a = pr1 (w,y,u) & b = pr2 (w,y,u) ) ) assume that A1: Gen w,y and A2: u = (a * w) + (b * y) ; ::_thesis: ( a = pr1 (w,y,u) & b = pr2 (w,y,u) ) u = ((pr1 (w,y,u)) * w) + ((pr2 (w,y,u)) * y) by A1, Lm15; hence ( a = pr1 (w,y,u) & b = pr2 (w,y,u) ) by A1, A2, Lm14; ::_thesis: verum end; Lm17: for V being RealLinearSpace for w, y, u, v being VECTOR of V for a being Real st Gen w,y holds ( pr1 (w,y,(u + v)) = (pr1 (w,y,u)) + (pr1 (w,y,v)) & pr2 (w,y,(u + v)) = (pr2 (w,y,u)) + (pr2 (w,y,v)) & pr1 (w,y,(u - v)) = (pr1 (w,y,u)) - (pr1 (w,y,v)) & pr2 (w,y,(u - v)) = (pr2 (w,y,u)) - (pr2 (w,y,v)) & pr1 (w,y,(a * u)) = a * (pr1 (w,y,u)) & pr2 (w,y,(a * u)) = a * (pr2 (w,y,u)) ) proof let V be RealLinearSpace; ::_thesis: for w, y, u, v being VECTOR of V for a being Real st Gen w,y holds ( pr1 (w,y,(u + v)) = (pr1 (w,y,u)) + (pr1 (w,y,v)) & pr2 (w,y,(u + v)) = (pr2 (w,y,u)) + (pr2 (w,y,v)) & pr1 (w,y,(u - v)) = (pr1 (w,y,u)) - (pr1 (w,y,v)) & pr2 (w,y,(u - v)) = (pr2 (w,y,u)) - (pr2 (w,y,v)) & pr1 (w,y,(a * u)) = a * (pr1 (w,y,u)) & pr2 (w,y,(a * u)) = a * (pr2 (w,y,u)) ) let w, y, u, v be VECTOR of V; ::_thesis: for a being Real st Gen w,y holds ( pr1 (w,y,(u + v)) = (pr1 (w,y,u)) + (pr1 (w,y,v)) & pr2 (w,y,(u + v)) = (pr2 (w,y,u)) + (pr2 (w,y,v)) & pr1 (w,y,(u - v)) = (pr1 (w,y,u)) - (pr1 (w,y,v)) & pr2 (w,y,(u - v)) = (pr2 (w,y,u)) - (pr2 (w,y,v)) & pr1 (w,y,(a * u)) = a * (pr1 (w,y,u)) & pr2 (w,y,(a * u)) = a * (pr2 (w,y,u)) ) let a be Real; ::_thesis: ( Gen w,y implies ( pr1 (w,y,(u + v)) = (pr1 (w,y,u)) + (pr1 (w,y,v)) & pr2 (w,y,(u + v)) = (pr2 (w,y,u)) + (pr2 (w,y,v)) & pr1 (w,y,(u - v)) = (pr1 (w,y,u)) - (pr1 (w,y,v)) & pr2 (w,y,(u - v)) = (pr2 (w,y,u)) - (pr2 (w,y,v)) & pr1 (w,y,(a * u)) = a * (pr1 (w,y,u)) & pr2 (w,y,(a * u)) = a * (pr2 (w,y,u)) ) ) set p1u = pr1 (w,y,u); set p2u = pr2 (w,y,u); set p1v = pr1 (w,y,v); set p2v = pr2 (w,y,v); assume A1: Gen w,y ; ::_thesis: ( pr1 (w,y,(u + v)) = (pr1 (w,y,u)) + (pr1 (w,y,v)) & pr2 (w,y,(u + v)) = (pr2 (w,y,u)) + (pr2 (w,y,v)) & pr1 (w,y,(u - v)) = (pr1 (w,y,u)) - (pr1 (w,y,v)) & pr2 (w,y,(u - v)) = (pr2 (w,y,u)) - (pr2 (w,y,v)) & pr1 (w,y,(a * u)) = a * (pr1 (w,y,u)) & pr2 (w,y,(a * u)) = a * (pr2 (w,y,u)) ) then A2: u = ((pr1 (w,y,u)) * w) + ((pr2 (w,y,u)) * y) by Lm15; A3: v = ((pr1 (w,y,v)) * w) + ((pr2 (w,y,v)) * y) by A1, Lm15; then u + v = ((((pr1 (w,y,u)) * w) + ((pr2 (w,y,u)) * y)) + ((pr1 (w,y,v)) * w)) + ((pr2 (w,y,v)) * y) by A2, RLVECT_1:def_3 .= ((((pr1 (w,y,u)) * w) + ((pr1 (w,y,v)) * w)) + ((pr2 (w,y,u)) * y)) + ((pr2 (w,y,v)) * y) by RLVECT_1:def_3 .= (((pr1 (w,y,u)) * w) + ((pr1 (w,y,v)) * w)) + (((pr2 (w,y,u)) * y) + ((pr2 (w,y,v)) * y)) by RLVECT_1:def_3 .= (((pr1 (w,y,u)) + (pr1 (w,y,v))) * w) + (((pr2 (w,y,u)) * y) + ((pr2 (w,y,v)) * y)) by RLVECT_1:def_6 .= (((pr1 (w,y,u)) + (pr1 (w,y,v))) * w) + (((pr2 (w,y,u)) + (pr2 (w,y,v))) * y) by RLVECT_1:def_6 ; hence ( pr1 (w,y,(u + v)) = (pr1 (w,y,u)) + (pr1 (w,y,v)) & pr2 (w,y,(u + v)) = (pr2 (w,y,u)) + (pr2 (w,y,v)) ) by A1, Lm16; ::_thesis: ( pr1 (w,y,(u - v)) = (pr1 (w,y,u)) - (pr1 (w,y,v)) & pr2 (w,y,(u - v)) = (pr2 (w,y,u)) - (pr2 (w,y,v)) & pr1 (w,y,(a * u)) = a * (pr1 (w,y,u)) & pr2 (w,y,(a * u)) = a * (pr2 (w,y,u)) ) u - v = (((pr1 (w,y,u)) - (pr1 (w,y,v))) * w) + (((pr2 (w,y,u)) - (pr2 (w,y,v))) * y) by A2, A3, Lm12; hence ( pr1 (w,y,(u - v)) = (pr1 (w,y,u)) - (pr1 (w,y,v)) & pr2 (w,y,(u - v)) = (pr2 (w,y,u)) - (pr2 (w,y,v)) ) by A1, Lm16; ::_thesis: ( pr1 (w,y,(a * u)) = a * (pr1 (w,y,u)) & pr2 (w,y,(a * u)) = a * (pr2 (w,y,u)) ) a * u = ((a * (pr1 (w,y,u))) * w) + ((a * (pr2 (w,y,u))) * y) by A2, Lm13; hence ( pr1 (w,y,(a * u)) = a * (pr1 (w,y,u)) & pr2 (w,y,(a * u)) = a * (pr2 (w,y,u)) ) by A1, Lm16; ::_thesis: verum end; Lm18: for V being RealLinearSpace for w, y, u, v being VECTOR of V st Gen w,y holds ( 2 * (pr1 (w,y,(u # v))) = (pr1 (w,y,u)) + (pr1 (w,y,v)) & 2 * (pr2 (w,y,(u # v))) = (pr2 (w,y,u)) + (pr2 (w,y,v)) ) proof let V be RealLinearSpace; ::_thesis: for w, y, u, v being VECTOR of V st Gen w,y holds ( 2 * (pr1 (w,y,(u # v))) = (pr1 (w,y,u)) + (pr1 (w,y,v)) & 2 * (pr2 (w,y,(u # v))) = (pr2 (w,y,u)) + (pr2 (w,y,v)) ) let w, y, u, v be VECTOR of V; ::_thesis: ( Gen w,y implies ( 2 * (pr1 (w,y,(u # v))) = (pr1 (w,y,u)) + (pr1 (w,y,v)) & 2 * (pr2 (w,y,(u # v))) = (pr2 (w,y,u)) + (pr2 (w,y,v)) ) ) assume A1: Gen w,y ; ::_thesis: ( 2 * (pr1 (w,y,(u # v))) = (pr1 (w,y,u)) + (pr1 (w,y,v)) & 2 * (pr2 (w,y,(u # v))) = (pr2 (w,y,u)) + (pr2 (w,y,v)) ) set p = u # v; 2 * (u # v) = (1 + 1) * (u # v) .= (1 * (u # v)) + (1 * (u # v)) by RLVECT_1:def_6 .= (u # v) + (1 * (u # v)) by RLVECT_1:def_8 .= (u # v) + (u # v) by RLVECT_1:def_8 .= u + v by Def2 ; then ( 2 * (pr1 (w,y,(u # v))) = pr1 (w,y,(u + v)) & 2 * (pr2 (w,y,(u # v))) = pr2 (w,y,(u + v)) ) by A1, Lm17; hence ( 2 * (pr1 (w,y,(u # v))) = (pr1 (w,y,u)) + (pr1 (w,y,v)) & 2 * (pr2 (w,y,(u # v))) = (pr2 (w,y,u)) + (pr2 (w,y,v)) ) by A1, Lm17; ::_thesis: verum end; Lm19: for V being RealLinearSpace for u, v being VECTOR of V holds (- u) + (- v) = - (u + v) proof let V be RealLinearSpace; ::_thesis: for u, v being VECTOR of V holds (- u) + (- v) = - (u + v) let u, v be VECTOR of V; ::_thesis: (- u) + (- v) = - (u + v) (u + v) + ((- u) + (- v)) = ((u + v) + (- u)) + (- v) by RLVECT_1:def_3 .= (v + (u + (- u))) + (- v) by RLVECT_1:def_3 .= (v + (0. V)) + (- v) by RLVECT_1:5 .= v + (- v) by RLVECT_1:4 .= 0. V by RLVECT_1:5 ; hence (- u) + (- v) = - (u + v) by RLVECT_1:def_10; ::_thesis: verum end; Lm20: for V being RealLinearSpace for u2, v, u1 being VECTOR of V for a2, a1 being Real holds (u2 + (a2 * v)) - (u1 + (a1 * v)) = (u2 - u1) + ((a2 - a1) * v) proof let V be RealLinearSpace; ::_thesis: for u2, v, u1 being VECTOR of V for a2, a1 being Real holds (u2 + (a2 * v)) - (u1 + (a1 * v)) = (u2 - u1) + ((a2 - a1) * v) let u2, v, u1 be VECTOR of V; ::_thesis: for a2, a1 being Real holds (u2 + (a2 * v)) - (u1 + (a1 * v)) = (u2 - u1) + ((a2 - a1) * v) let a2, a1 be Real; ::_thesis: (u2 + (a2 * v)) - (u1 + (a1 * v)) = (u2 - u1) + ((a2 - a1) * v) thus (u2 + (a2 * v)) - (u1 + (a1 * v)) = ((u2 + (a2 * v)) - u1) - (a1 * v) by RLVECT_1:27 .= ((a2 * v) + (u2 - u1)) - (a1 * v) by RLVECT_1:def_3 .= (u2 - u1) + ((a2 * v) - (a1 * v)) by RLVECT_1:def_3 .= (u2 - u1) + ((a2 - a1) * v) by RLVECT_1:35 ; ::_thesis: verum end; definition let V be RealLinearSpace; let w, y, u, v be VECTOR of V; func PProJ (w,y,u,v) -> Real equals :: GEOMTRAP:def 6 ((pr1 (w,y,u)) * (pr1 (w,y,v))) + ((pr2 (w,y,u)) * (pr2 (w,y,v))); correctness coherence ((pr1 (w,y,u)) * (pr1 (w,y,v))) + ((pr2 (w,y,u)) * (pr2 (w,y,v))) is Real; ; end; :: deftheorem defines PProJ GEOMTRAP:def_6_:_ for V being RealLinearSpace for w, y, u, v being VECTOR of V holds PProJ (w,y,u,v) = ((pr1 (w,y,u)) * (pr1 (w,y,v))) + ((pr2 (w,y,u)) * (pr2 (w,y,v))); theorem :: GEOMTRAP:29 for V being RealLinearSpace for w, y, u, v being VECTOR of V holds PProJ (w,y,u,v) = PProJ (w,y,v,u) ; theorem Th30: :: GEOMTRAP:30 for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for u, v, v1 being VECTOR of V holds ( PProJ (w,y,u,(v + v1)) = (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) & PProJ (w,y,u,(v - v1)) = (PProJ (w,y,u,v)) - (PProJ (w,y,u,v1)) & PProJ (w,y,(v - v1),u) = (PProJ (w,y,v,u)) - (PProJ (w,y,v1,u)) & PProJ (w,y,(v + v1),u) = (PProJ (w,y,v,u)) + (PProJ (w,y,v1,u)) ) proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V st Gen w,y holds for u, v, v1 being VECTOR of V holds ( PProJ (w,y,u,(v + v1)) = (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) & PProJ (w,y,u,(v - v1)) = (PProJ (w,y,u,v)) - (PProJ (w,y,u,v1)) & PProJ (w,y,(v - v1),u) = (PProJ (w,y,v,u)) - (PProJ (w,y,v1,u)) & PProJ (w,y,(v + v1),u) = (PProJ (w,y,v,u)) + (PProJ (w,y,v1,u)) ) let w, y be VECTOR of V; ::_thesis: ( Gen w,y implies for u, v, v1 being VECTOR of V holds ( PProJ (w,y,u,(v + v1)) = (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) & PProJ (w,y,u,(v - v1)) = (PProJ (w,y,u,v)) - (PProJ (w,y,u,v1)) & PProJ (w,y,(v - v1),u) = (PProJ (w,y,v,u)) - (PProJ (w,y,v1,u)) & PProJ (w,y,(v + v1),u) = (PProJ (w,y,v,u)) + (PProJ (w,y,v1,u)) ) ) assume A1: Gen w,y ; ::_thesis: for u, v, v1 being VECTOR of V holds ( PProJ (w,y,u,(v + v1)) = (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) & PProJ (w,y,u,(v - v1)) = (PProJ (w,y,u,v)) - (PProJ (w,y,u,v1)) & PProJ (w,y,(v - v1),u) = (PProJ (w,y,v,u)) - (PProJ (w,y,v1,u)) & PProJ (w,y,(v + v1),u) = (PProJ (w,y,v,u)) + (PProJ (w,y,v1,u)) ) let u, v, v1 be VECTOR of V; ::_thesis: ( PProJ (w,y,u,(v + v1)) = (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) & PProJ (w,y,u,(v - v1)) = (PProJ (w,y,u,v)) - (PProJ (w,y,u,v1)) & PProJ (w,y,(v - v1),u) = (PProJ (w,y,v,u)) - (PProJ (w,y,v1,u)) & PProJ (w,y,(v + v1),u) = (PProJ (w,y,v,u)) + (PProJ (w,y,v1,u)) ) A2: now__::_thesis:_for_u,_v,_v1_being_VECTOR_of_V_holds_ (_PProJ_(w,y,u,(v_+_v1))_=_(PProJ_(w,y,u,v))_+_(PProJ_(w,y,u,v1))_&_PProJ_(w,y,u,(v_-_v1))_=_(PProJ_(w,y,u,v))_-_(PProJ_(w,y,u,v1))_) let u, v, v1 be VECTOR of V; ::_thesis: ( PProJ (w,y,u,(v + v1)) = (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) & PProJ (w,y,u,(v - v1)) = (PProJ (w,y,u,v)) - (PProJ (w,y,u,v1)) ) A3: PProJ (w,y,u,(v - v1)) = ((pr1 (w,y,u)) * ((pr1 (w,y,v)) - (pr1 (w,y,v1)))) + ((pr2 (w,y,u)) * (pr2 (w,y,(v - v1)))) by A1, Lm17 .= ((pr1 (w,y,u)) * ((pr1 (w,y,v)) - (pr1 (w,y,v1)))) + ((pr2 (w,y,u)) * ((pr2 (w,y,v)) - (pr2 (w,y,v1)))) by A1, Lm17 .= (PProJ (w,y,u,v)) - (PProJ (w,y,u,v1)) ; PProJ (w,y,u,(v + v1)) = ((pr1 (w,y,u)) * ((pr1 (w,y,v)) + (pr1 (w,y,v1)))) + ((pr2 (w,y,u)) * (pr2 (w,y,(v + v1)))) by A1, Lm17 .= ((pr1 (w,y,u)) * ((pr1 (w,y,v)) + (pr1 (w,y,v1)))) + ((pr2 (w,y,u)) * ((pr2 (w,y,v)) + (pr2 (w,y,v1)))) by A1, Lm17 .= (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) ; hence ( PProJ (w,y,u,(v + v1)) = (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) & PProJ (w,y,u,(v - v1)) = (PProJ (w,y,u,v)) - (PProJ (w,y,u,v1)) ) by A3; ::_thesis: verum end; hence ( PProJ (w,y,u,(v + v1)) = (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) & PProJ (w,y,u,(v - v1)) = (PProJ (w,y,u,v)) - (PProJ (w,y,u,v1)) ) ; ::_thesis: ( PProJ (w,y,(v - v1),u) = (PProJ (w,y,v,u)) - (PProJ (w,y,v1,u)) & PProJ (w,y,(v + v1),u) = (PProJ (w,y,v,u)) + (PProJ (w,y,v1,u)) ) thus PProJ (w,y,(v - v1),u) = PProJ (w,y,u,(v - v1)) .= (PProJ (w,y,u,v)) - (PProJ (w,y,u,v1)) by A2 .= (PProJ (w,y,v,u)) - (PProJ (w,y,v1,u)) ; ::_thesis: PProJ (w,y,(v + v1),u) = (PProJ (w,y,v,u)) + (PProJ (w,y,v1,u)) thus PProJ (w,y,(v + v1),u) = PProJ (w,y,u,(v + v1)) .= (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) by A2 .= (PProJ (w,y,v,u)) + (PProJ (w,y,v1,u)) ; ::_thesis: verum end; theorem Th31: :: GEOMTRAP:31 for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for u, v being VECTOR of V for a being Real holds ( PProJ (w,y,(a * u),v) = a * (PProJ (w,y,u,v)) & PProJ (w,y,u,(a * v)) = a * (PProJ (w,y,u,v)) & PProJ (w,y,(a * u),v) = (PProJ (w,y,u,v)) * a & PProJ (w,y,u,(a * v)) = (PProJ (w,y,u,v)) * a ) proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V st Gen w,y holds for u, v being VECTOR of V for a being Real holds ( PProJ (w,y,(a * u),v) = a * (PProJ (w,y,u,v)) & PProJ (w,y,u,(a * v)) = a * (PProJ (w,y,u,v)) & PProJ (w,y,(a * u),v) = (PProJ (w,y,u,v)) * a & PProJ (w,y,u,(a * v)) = (PProJ (w,y,u,v)) * a ) let w, y be VECTOR of V; ::_thesis: ( Gen w,y implies for u, v being VECTOR of V for a being Real holds ( PProJ (w,y,(a * u),v) = a * (PProJ (w,y,u,v)) & PProJ (w,y,u,(a * v)) = a * (PProJ (w,y,u,v)) & PProJ (w,y,(a * u),v) = (PProJ (w,y,u,v)) * a & PProJ (w,y,u,(a * v)) = (PProJ (w,y,u,v)) * a ) ) assume A1: Gen w,y ; ::_thesis: for u, v being VECTOR of V for a being Real holds ( PProJ (w,y,(a * u),v) = a * (PProJ (w,y,u,v)) & PProJ (w,y,u,(a * v)) = a * (PProJ (w,y,u,v)) & PProJ (w,y,(a * u),v) = (PProJ (w,y,u,v)) * a & PProJ (w,y,u,(a * v)) = (PProJ (w,y,u,v)) * a ) A2: now__::_thesis:_for_u,_v_being_VECTOR_of_V for_a_being_Real_holds_PProJ_(w,y,(a_*_u),v)_=_a_*_(PProJ_(w,y,u,v)) let u, v be VECTOR of V; ::_thesis: for a being Real holds PProJ (w,y,(a * u),v) = a * (PProJ (w,y,u,v)) let a be Real; ::_thesis: PProJ (w,y,(a * u),v) = a * (PProJ (w,y,u,v)) PProJ (w,y,(a * u),v) = ((a * (pr1 (w,y,u))) * (pr1 (w,y,v))) + ((pr2 (w,y,(a * u))) * (pr2 (w,y,v))) by A1, Lm17 .= ((a * (pr1 (w,y,u))) * (pr1 (w,y,v))) + ((a * (pr2 (w,y,u))) * (pr2 (w,y,v))) by A1, Lm17 .= a * (PProJ (w,y,u,v)) ; hence PProJ (w,y,(a * u),v) = a * (PProJ (w,y,u,v)) ; ::_thesis: verum end; let u, v be VECTOR of V; ::_thesis: for a being Real holds ( PProJ (w,y,(a * u),v) = a * (PProJ (w,y,u,v)) & PProJ (w,y,u,(a * v)) = a * (PProJ (w,y,u,v)) & PProJ (w,y,(a * u),v) = (PProJ (w,y,u,v)) * a & PProJ (w,y,u,(a * v)) = (PProJ (w,y,u,v)) * a ) let a be Real; ::_thesis: ( PProJ (w,y,(a * u),v) = a * (PProJ (w,y,u,v)) & PProJ (w,y,u,(a * v)) = a * (PProJ (w,y,u,v)) & PProJ (w,y,(a * u),v) = (PProJ (w,y,u,v)) * a & PProJ (w,y,u,(a * v)) = (PProJ (w,y,u,v)) * a ) thus PProJ (w,y,(a * u),v) = a * (PProJ (w,y,u,v)) by A2; ::_thesis: ( PProJ (w,y,u,(a * v)) = a * (PProJ (w,y,u,v)) & PProJ (w,y,(a * u),v) = (PProJ (w,y,u,v)) * a & PProJ (w,y,u,(a * v)) = (PProJ (w,y,u,v)) * a ) thus PProJ (w,y,u,(a * v)) = PProJ (w,y,(a * v),u) .= a * (PProJ (w,y,v,u)) by A2 .= a * (PProJ (w,y,u,v)) ; ::_thesis: ( PProJ (w,y,(a * u),v) = (PProJ (w,y,u,v)) * a & PProJ (w,y,u,(a * v)) = (PProJ (w,y,u,v)) * a ) thus PProJ (w,y,(a * u),v) = (PProJ (w,y,u,v)) * a by A2; ::_thesis: PProJ (w,y,u,(a * v)) = (PProJ (w,y,u,v)) * a thus PProJ (w,y,u,(a * v)) = PProJ (w,y,(a * v),u) .= a * (PProJ (w,y,v,u)) by A2 .= (PProJ (w,y,u,v)) * a ; ::_thesis: verum end; theorem Th32: :: GEOMTRAP:32 for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for u, v being VECTOR of V holds ( u,v are_Ort_wrt w,y iff PProJ (w,y,u,v) = 0 ) proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V st Gen w,y holds for u, v being VECTOR of V holds ( u,v are_Ort_wrt w,y iff PProJ (w,y,u,v) = 0 ) let w, y be VECTOR of V; ::_thesis: ( Gen w,y implies for u, v being VECTOR of V holds ( u,v are_Ort_wrt w,y iff PProJ (w,y,u,v) = 0 ) ) assume A1: Gen w,y ; ::_thesis: for u, v being VECTOR of V holds ( u,v are_Ort_wrt w,y iff PProJ (w,y,u,v) = 0 ) let u, v be VECTOR of V; ::_thesis: ( u,v are_Ort_wrt w,y iff PProJ (w,y,u,v) = 0 ) set a1 = pr1 (w,y,u); set a2 = pr2 (w,y,u); set b1 = pr1 (w,y,v); set b2 = pr2 (w,y,v); ( u = ((pr1 (w,y,u)) * w) + ((pr2 (w,y,u)) * y) & v = ((pr1 (w,y,v)) * w) + ((pr2 (w,y,v)) * y) ) by A1, Lm15; hence ( u,v are_Ort_wrt w,y iff PProJ (w,y,u,v) = 0 ) by A1, ANALMETR:1, ANALMETR:def_2; ::_thesis: verum end; theorem Th33: :: GEOMTRAP:33 for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for u, v, u1, v1 being VECTOR of V holds ( u,v,u1,v1 are_Ort_wrt w,y iff PProJ (w,y,(v - u),(v1 - u1)) = 0 ) proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V st Gen w,y holds for u, v, u1, v1 being VECTOR of V holds ( u,v,u1,v1 are_Ort_wrt w,y iff PProJ (w,y,(v - u),(v1 - u1)) = 0 ) let w, y be VECTOR of V; ::_thesis: ( Gen w,y implies for u, v, u1, v1 being VECTOR of V holds ( u,v,u1,v1 are_Ort_wrt w,y iff PProJ (w,y,(v - u),(v1 - u1)) = 0 ) ) assume A1: Gen w,y ; ::_thesis: for u, v, u1, v1 being VECTOR of V holds ( u,v,u1,v1 are_Ort_wrt w,y iff PProJ (w,y,(v - u),(v1 - u1)) = 0 ) let u, v, u1, v1 be VECTOR of V; ::_thesis: ( u,v,u1,v1 are_Ort_wrt w,y iff PProJ (w,y,(v - u),(v1 - u1)) = 0 ) ( u,v,u1,v1 are_Ort_wrt w,y iff v - u,v1 - u1 are_Ort_wrt w,y ) by ANALMETR:def_3; hence ( u,v,u1,v1 are_Ort_wrt w,y iff PProJ (w,y,(v - u),(v1 - u1)) = 0 ) by A1, Th32; ::_thesis: verum end; theorem Th34: :: GEOMTRAP:34 for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for u, v, v1 being VECTOR of V holds 2 * (PProJ (w,y,u,(v # v1))) = (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V st Gen w,y holds for u, v, v1 being VECTOR of V holds 2 * (PProJ (w,y,u,(v # v1))) = (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) let w, y be VECTOR of V; ::_thesis: ( Gen w,y implies for u, v, v1 being VECTOR of V holds 2 * (PProJ (w,y,u,(v # v1))) = (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) ) assume A1: Gen w,y ; ::_thesis: for u, v, v1 being VECTOR of V holds 2 * (PProJ (w,y,u,(v # v1))) = (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) let u, v, v1 be VECTOR of V; ::_thesis: 2 * (PProJ (w,y,u,(v # v1))) = (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) set a1 = pr1 (w,y,u); set a2 = pr2 (w,y,u); set b1 = pr1 (w,y,v); set b2 = pr2 (w,y,v); set c1 = pr1 (w,y,v1); set c2 = pr2 (w,y,v1); thus 2 * (PProJ (w,y,u,(v # v1))) = ((pr1 (w,y,u)) * (2 * (pr1 (w,y,(v # v1))))) + (2 * ((pr2 (w,y,u)) * (pr2 (w,y,(v # v1))))) .= ((pr1 (w,y,u)) * ((pr1 (w,y,v)) + (pr1 (w,y,v1)))) + ((pr2 (w,y,u)) * (2 * (pr2 (w,y,(v # v1))))) by A1, Lm18 .= ((pr1 (w,y,u)) * ((pr1 (w,y,v)) + (pr1 (w,y,v1)))) + ((pr2 (w,y,u)) * ((pr2 (w,y,v)) + (pr2 (w,y,v1)))) by A1, Lm18 .= (PProJ (w,y,u,v)) + (PProJ (w,y,u,v1)) ; ::_thesis: verum end; theorem Th35: :: GEOMTRAP:35 for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for u, v being VECTOR of V st u <> v holds PProJ (w,y,(u - v),(u - v)) <> 0 proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V st Gen w,y holds for u, v being VECTOR of V st u <> v holds PProJ (w,y,(u - v),(u - v)) <> 0 let w, y be VECTOR of V; ::_thesis: ( Gen w,y implies for u, v being VECTOR of V st u <> v holds PProJ (w,y,(u - v),(u - v)) <> 0 ) assume A1: Gen w,y ; ::_thesis: for u, v being VECTOR of V st u <> v holds PProJ (w,y,(u - v),(u - v)) <> 0 let u, v be VECTOR of V; ::_thesis: ( u <> v implies PProJ (w,y,(u - v),(u - v)) <> 0 ) assume A2: u <> v ; ::_thesis: PProJ (w,y,(u - v),(u - v)) <> 0 assume PProJ (w,y,(u - v),(u - v)) = 0 ; ::_thesis: contradiction then u - v,u - v are_Ort_wrt w,y by A1, Th32; then u - v = 0. V by A1, ANALMETR:11; hence contradiction by A2, RLVECT_1:21; ::_thesis: verum end; theorem Th36: :: GEOMTRAP:36 for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for p, q, u, v, v9 being VECTOR of V for A being Real st p,q,u,v are_DTr_wrt w,y & p <> q & A = ((PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u)))) * ((PProJ (w,y,(p - q),(p - q))) ") & v9 = u + (A * (p - q)) holds v = v9 proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V st Gen w,y holds for p, q, u, v, v9 being VECTOR of V for A being Real st p,q,u,v are_DTr_wrt w,y & p <> q & A = ((PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u)))) * ((PProJ (w,y,(p - q),(p - q))) ") & v9 = u + (A * (p - q)) holds v = v9 let w, y be VECTOR of V; ::_thesis: ( Gen w,y implies for p, q, u, v, v9 being VECTOR of V for A being Real st p,q,u,v are_DTr_wrt w,y & p <> q & A = ((PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u)))) * ((PProJ (w,y,(p - q),(p - q))) ") & v9 = u + (A * (p - q)) holds v = v9 ) assume A1: Gen w,y ; ::_thesis: for p, q, u, v, v9 being VECTOR of V for A being Real st p,q,u,v are_DTr_wrt w,y & p <> q & A = ((PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u)))) * ((PProJ (w,y,(p - q),(p - q))) ") & v9 = u + (A * (p - q)) holds v = v9 let p, q, u, v, v9 be VECTOR of V; ::_thesis: for A being Real st p,q,u,v are_DTr_wrt w,y & p <> q & A = ((PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u)))) * ((PProJ (w,y,(p - q),(p - q))) ") & v9 = u + (A * (p - q)) holds v = v9 let A be Real; ::_thesis: ( p,q,u,v are_DTr_wrt w,y & p <> q & A = ((PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u)))) * ((PProJ (w,y,(p - q),(p - q))) ") & v9 = u + (A * (p - q)) implies v = v9 ) assume that A2: p,q,u,v are_DTr_wrt w,y and A3: p <> q and A4: A = ((PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u)))) * ((PProJ (w,y,(p - q),(p - q))) ") and A5: v9 = u + (A * (p - q)) ; ::_thesis: v = v9 A6: PProJ (w,y,(p - q),(p - q)) <> 0 by A1, A3, Th35; A7: PProJ (w,y,(p - q),(A * (p - q))) = (((PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u)))) * ((PProJ (w,y,(p - q),(p - q))) ")) * (PProJ (w,y,(p - q),(p - q))) by A1, A4, Th31 .= ((PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u)))) * (((PProJ (w,y,(p - q),(p - q))) ") * (PProJ (w,y,(p - q),(p - q)))) .= ((PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u)))) * 1 by A6, XCMPLX_0:def_7 .= (PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u))) ; set s = p # q; set X = PProJ (w,y,(p - q),((v9 # u) - (p # q))); 2 * (PProJ (w,y,(p - q),((v9 # u) - (p # q)))) = 2 * ((PProJ (w,y,(p - q),(v9 # u))) - (PProJ (w,y,(p - q),(p # q)))) by A1, Th30 .= (2 * (PProJ (w,y,(p - q),(v9 # u)))) - (2 * (PProJ (w,y,(p - q),(p # q)))) .= ((PProJ (w,y,(p - q),v9)) + (PProJ (w,y,(p - q),u))) - (2 * (PProJ (w,y,(p - q),(p # q)))) by A1, Th34 .= ((PProJ (w,y,(p - q),v9)) + (PProJ (w,y,(p - q),u))) - ((PProJ (w,y,(p - q),p)) + (PProJ (w,y,(p - q),q))) by A1, Th34 .= ((PProJ (w,y,(p - q),(u + (A * (p - q))))) + (PProJ (w,y,(p - q),u))) - (PProJ (w,y,(p - q),(p + q))) by A1, A5, Th30 .= (((PProJ (w,y,(p - q),u)) + (PProJ (w,y,(p - q),(A * (p - q))))) + (PProJ (w,y,(p - q),u))) - (PProJ (w,y,(p - q),(p + q))) by A1, Th30 ; then q,p,p # q,v9 # u are_Ort_wrt w,y by A1, A7, Th33; then p # q,v9 # u,q,p are_Ort_wrt w,y by Lm5; then A8: p # q,v9 # u,p,q are_Ort_wrt w,y by Lm4; set Y = PProJ (w,y,(v9 - u),((v9 # u) - (p # q))); A9: v9 - u = A * (p - q) by A5, RLSUB_2:61; 1 * (v9 - u) = (u + (A * (p - q))) - u by A5, RLVECT_1:def_8 .= A * (p - q) by RLSUB_2:61 ; then ( q,p // u,v9 or q,p // v9,u ) by ANALMETR:14; then A10: ( p,q // u,v9 or p,q // v9,u ) by ANALOAF:12; A11: PProJ (w,y,(A * (p - q)),(A * (p - q))) = A * (PProJ (w,y,(p - q),(A * (p - q)))) by A1, Th31 .= A * ((((PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u)))) * ((PProJ (w,y,(p - q),(p - q))) ")) * (PProJ (w,y,(p - q),(p - q)))) by A1, A4, Th31 .= A * (((PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u)))) * (((PProJ (w,y,(p - q),(p - q))) ") * (PProJ (w,y,(p - q),(p - q))))) .= A * (((PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u)))) * 1) by A6, XCMPLX_0:def_7 .= A * ((PProJ (w,y,(p - q),(p + q))) - (2 * (PProJ (w,y,(p - q),u)))) ; 2 * (PProJ (w,y,(v9 - u),((v9 # u) - (p # q)))) = 2 * ((PProJ (w,y,(v9 - u),(v9 # u))) - (PProJ (w,y,(v9 - u),(p # q)))) by A1, Th30 .= (2 * (PProJ (w,y,(v9 - u),(v9 # u)))) - (2 * (PProJ (w,y,(v9 - u),(p # q)))) .= ((PProJ (w,y,(v9 - u),v9)) + (PProJ (w,y,(v9 - u),u))) - (2 * (PProJ (w,y,(v9 - u),(p # q)))) by A1, Th34 .= ((PProJ (w,y,(v9 - u),v9)) + (PProJ (w,y,(v9 - u),u))) - ((PProJ (w,y,(v9 - u),p)) + (PProJ (w,y,(v9 - u),q))) by A1, Th34 .= ((PProJ (w,y,(v9 - u),(u + (A * (p - q))))) + (PProJ (w,y,(v9 - u),u))) - (PProJ (w,y,(v9 - u),(p + q))) by A1, A5, Th30 .= (((PProJ (w,y,(v9 - u),u)) + (PProJ (w,y,(v9 - u),(A * (p - q))))) + (PProJ (w,y,(v9 - u),u))) - (PProJ (w,y,(v9 - u),(p + q))) by A1, Th30 ; then A12: 2 * (PProJ (w,y,(v9 - u),((v9 # u) - (p # q)))) = (((PProJ (w,y,(A * (p - q)),u)) + ((A * (PProJ (w,y,(p - q),(p + q)))) - (A * (2 * (PProJ (w,y,(p - q),u)))))) + (PProJ (w,y,(A * (p - q)),u))) - (PProJ (w,y,(A * (p - q)),(p + q))) by A9, A11 .= (((PProJ (w,y,(A * (p - q)),u)) + ((PProJ (w,y,(A * (p - q)),(p + q))) - (2 * (A * (PProJ (w,y,(p - q),u)))))) + (PProJ (w,y,(A * (p - q)),u))) - (PProJ (w,y,(A * (p - q)),(p + q))) by A1, Th31 .= (((PProJ (w,y,(A * (p - q)),u)) + ((PProJ (w,y,(A * (p - q)),(p + q))) - (2 * (PProJ (w,y,(A * (p - q)),u))))) + (PProJ (w,y,(A * (p - q)),u))) - (PProJ (w,y,(A * (p - q)),(p + q))) by A1, Th31 .= 0 ; then u,v9,p # q,v9 # u are_Ort_wrt w,y by A1, Th33; then p # q,v9 # u,u,v9 are_Ort_wrt w,y by Lm5; then p # q,v9 # u,v9,u are_Ort_wrt w,y by Lm4; then ( p,q,p # q,u # v9 are_Ort_wrt w,y & u,v9,p # q,u # v9 are_Ort_wrt w,y & p,q,p # q,v9 # u are_Ort_wrt w,y & v9,u,p # q,v9 # u are_Ort_wrt w,y ) by A1, A8, A12, Lm5, Th33; then ( p,q,u,v9 are_DTr_wrt w,y or p,q,v9,u are_DTr_wrt w,y ) by A10, Def3; hence v = v9 by A1, A2, A3, Th25; ::_thesis: verum end; Lm21: for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for u, u9, u1, u2, t1, t2 being VECTOR of V for A1, A2 being Real st A1 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") & A2 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") & t1 = u1 + (A1 * (u - u9)) & t2 = u2 + (A2 * (u - u9)) holds ( t2 - t1 = (u2 - u1) + ((A2 - A1) * (u - u9)) & A2 - A1 = ((- 2) * (PProJ (w,y,(u - u9),(u2 - u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") ) proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V st Gen w,y holds for u, u9, u1, u2, t1, t2 being VECTOR of V for A1, A2 being Real st A1 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") & A2 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") & t1 = u1 + (A1 * (u - u9)) & t2 = u2 + (A2 * (u - u9)) holds ( t2 - t1 = (u2 - u1) + ((A2 - A1) * (u - u9)) & A2 - A1 = ((- 2) * (PProJ (w,y,(u - u9),(u2 - u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") ) let w, y be VECTOR of V; ::_thesis: ( Gen w,y implies for u, u9, u1, u2, t1, t2 being VECTOR of V for A1, A2 being Real st A1 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") & A2 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") & t1 = u1 + (A1 * (u - u9)) & t2 = u2 + (A2 * (u - u9)) holds ( t2 - t1 = (u2 - u1) + ((A2 - A1) * (u - u9)) & A2 - A1 = ((- 2) * (PProJ (w,y,(u - u9),(u2 - u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") ) ) assume A1: Gen w,y ; ::_thesis: for u, u9, u1, u2, t1, t2 being VECTOR of V for A1, A2 being Real st A1 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") & A2 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") & t1 = u1 + (A1 * (u - u9)) & t2 = u2 + (A2 * (u - u9)) holds ( t2 - t1 = (u2 - u1) + ((A2 - A1) * (u - u9)) & A2 - A1 = ((- 2) * (PProJ (w,y,(u - u9),(u2 - u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") ) let u, u9, u1, u2, t1, t2 be VECTOR of V; ::_thesis: for A1, A2 being Real st A1 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") & A2 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") & t1 = u1 + (A1 * (u - u9)) & t2 = u2 + (A2 * (u - u9)) holds ( t2 - t1 = (u2 - u1) + ((A2 - A1) * (u - u9)) & A2 - A1 = ((- 2) * (PProJ (w,y,(u - u9),(u2 - u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") ) let A1, A2 be Real; ::_thesis: ( A1 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") & A2 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") & t1 = u1 + (A1 * (u - u9)) & t2 = u2 + (A2 * (u - u9)) implies ( t2 - t1 = (u2 - u1) + ((A2 - A1) * (u - u9)) & A2 - A1 = ((- 2) * (PProJ (w,y,(u - u9),(u2 - u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") ) ) assume A2: ( A1 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") & A2 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") & t1 = u1 + (A1 * (u - u9)) & t2 = u2 + (A2 * (u - u9)) ) ; ::_thesis: ( t2 - t1 = (u2 - u1) + ((A2 - A1) * (u - u9)) & A2 - A1 = ((- 2) * (PProJ (w,y,(u - u9),(u2 - u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") ) thus ( t2 - t1 = (u2 - u1) + ((A2 - A1) * (u - u9)) & A2 - A1 = ((- 2) * (PProJ (w,y,(u - u9),(u2 - u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") ) ::_thesis: verum proof set uu = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) - ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))); A3: (2 * u1) - (2 * u2) = - ((2 * u2) - (2 * u1)) by RLVECT_1:33 .= - (2 * (u2 - u1)) by RLVECT_1:34 .= 2 * (- (u2 - u1)) by RLVECT_1:25 .= (- 2) * (u2 - u1) by RLVECT_1:24 ; ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) - ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) = ((PProJ (w,y,(u - u9),(u + u9))) - (PProJ (w,y,(u - u9),(2 * u2)))) - ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) by A1, Th31 .= ((PProJ (w,y,(u - u9),(u + u9))) - (PProJ (w,y,(u - u9),(2 * u2)))) - ((PProJ (w,y,(u - u9),(u + u9))) - (PProJ (w,y,(u - u9),(2 * u1)))) by A1, Th31 .= (PProJ (w,y,(u - u9),(2 * u1))) - (PProJ (w,y,(u - u9),(2 * u2))) .= PProJ (w,y,(u - u9),((2 * u1) - (2 * u2))) by A1, Th30 ; then ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) - ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) = (- 2) * (PProJ (w,y,(u - u9),(u2 - u1))) by A1, A3, Th31; hence ( t2 - t1 = (u2 - u1) + ((A2 - A1) * (u - u9)) & A2 - A1 = ((- 2) * (PProJ (w,y,(u - u9),(u2 - u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ") ) by A2, Lm20, XCMPLX_1:40; ::_thesis: verum end; end; theorem Th37: :: GEOMTRAP:37 for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for u, u9, u1, u2, v1, v2, t1, t2, w1, w2 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2 // v1,v2 holds t1,t2 // w1,w2 proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V st Gen w,y holds for u, u9, u1, u2, v1, v2, t1, t2, w1, w2 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2 // v1,v2 holds t1,t2 // w1,w2 let w, y be VECTOR of V; ::_thesis: ( Gen w,y implies for u, u9, u1, u2, v1, v2, t1, t2, w1, w2 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2 // v1,v2 holds t1,t2 // w1,w2 ) assume A1: Gen w,y ; ::_thesis: for u, u9, u1, u2, v1, v2, t1, t2, w1, w2 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2 // v1,v2 holds t1,t2 // w1,w2 let u, u9, u1, u2, v1, v2, t1, t2, w1, w2 be VECTOR of V; ::_thesis: ( u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2 // v1,v2 implies t1,t2 // w1,w2 ) assume that A2: u <> u9 and A3: ( u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y ) and A4: ( u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y ) and A5: u1,u2 // v1,v2 ; ::_thesis: t1,t2 // w1,w2 set A1 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "); set A2 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) "); set A3 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "); set A4 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) "); A6: ( u1 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) = t1 & u2 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) = t2 ) by A1, A2, A3, Th36; A7: ( v1 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) = w1 & v2 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) = w2 ) by A1, A2, A4, Th36; A8: ( t1 = u1 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) & t2 = u2 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) ) by A1, A2, A3, Th36; percases ( u1 = u2 or v1 = v2 or ( u1 <> u2 & v1 <> v2 ) ) ; suppose u1 = u2 ; ::_thesis: t1,t2 // w1,w2 then t1 = t2 by A1, A2, A3, Th24; hence t1,t2 // w1,w2 by ANALOAF:def_1; ::_thesis: verum end; suppose v1 = v2 ; ::_thesis: t1,t2 // w1,w2 then w1 = w2 by A1, A2, A4, Th24; hence t1,t2 // w1,w2 by ANALOAF:def_1; ::_thesis: verum end; supposeA9: ( u1 <> u2 & v1 <> v2 ) ; ::_thesis: t1,t2 // w1,w2 set cc = (PProJ (w,y,(u - u9),(u - u9))) " ; set vv = (- 2) * (PProJ (w,y,(u - u9),(v2 - v1))); set uu = (- 2) * (PProJ (w,y,(u - u9),(u2 - u1))); A10: w2 - w1 = (v2 - v1) + (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)) by A1, A7, Lm21; consider a, b being Real such that A11: a * (u2 - u1) = b * (v2 - v1) and A12: ( 0 < a & 0 < b ) by A5, A9, ANALOAF:7; A13: a * ((- 2) * (PProJ (w,y,(u - u9),(u2 - u1)))) = (- 2) * (a * (PProJ (w,y,(u - u9),(u2 - u1)))) .= (- 2) * (PProJ (w,y,(u - u9),(b * (v2 - v1)))) by A1, A11, Th31 .= (- 2) * (b * (PProJ (w,y,(u - u9),(v2 - v1)))) by A1, Th31 .= b * ((- 2) * (PProJ (w,y,(u - u9),(v2 - v1)))) ; A14: a * ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) = a * (((- 2) * (PProJ (w,y,(u - u9),(u2 - u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) by A1, A8, Lm21 .= (b * ((- 2) * (PProJ (w,y,(u - u9),(v2 - v1))))) * ((PProJ (w,y,(u - u9),(u - u9))) ") by A13, XCMPLX_1:4 .= b * (((- 2) * (PProJ (w,y,(u - u9),(v2 - v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) .= b * ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) by A1, A7, Lm21 ; t2 - t1 = (u2 - u1) + (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)) by A1, A6, Lm21; then a * (t2 - t1) = (a * (u2 - u1)) + (a * (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9))) by RLVECT_1:def_5 .= (b * (v2 - v1)) + ((b * ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")))) * (u - u9)) by A11, A14, RLVECT_1:def_7 .= (b * (v2 - v1)) + (b * (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9))) by RLVECT_1:def_7 .= b * (w2 - w1) by A10, RLVECT_1:def_5 ; hence t1,t2 // w1,w2 by A12, ANALOAF:def_1; ::_thesis: verum end; end; end; theorem :: GEOMTRAP:38 for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for u, u9, u1, u2, v1, t1, t2, w1 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & v1 = u1 # u2 holds w1 = t1 # t2 proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V st Gen w,y holds for u, u9, u1, u2, v1, t1, t2, w1 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & v1 = u1 # u2 holds w1 = t1 # t2 let w, y be VECTOR of V; ::_thesis: ( Gen w,y implies for u, u9, u1, u2, v1, t1, t2, w1 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & v1 = u1 # u2 holds w1 = t1 # t2 ) assume A1: Gen w,y ; ::_thesis: for u, u9, u1, u2, v1, t1, t2, w1 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & v1 = u1 # u2 holds w1 = t1 # t2 let u, u9, u1, u2, v1, t1, t2, w1 be VECTOR of V; ::_thesis: ( u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & v1 = u1 # u2 implies w1 = t1 # t2 ) assume that A2: u <> u9 and A3: ( u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y ) and A4: u,u9,v1,w1 are_DTr_wrt w,y and A5: v1 = u1 # u2 ; ::_thesis: w1 = t1 # t2 set G = PProJ (w,y,(u - u9),(u + u9)); set H = PProJ (w,y,(u - u9),u1); set W = PProJ (w,y,(u - u9),u2); set I = PProJ (w,y,(u - u9),v1); set N = PProJ (w,y,(u - u9),(u - u9)); set A1 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "); set A2 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) "); set A3 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "); A6: (PProJ (w,y,(u - u9),u1)) + (PProJ (w,y,(u - u9),u2)) = PProJ (w,y,(u - u9),(u1 + u2)) by A1, Th30 .= PProJ (w,y,(u - u9),(v1 + v1)) by A5, Def2 .= (PProJ (w,y,(u - u9),v1)) + (PProJ (w,y,(u - u9),v1)) by A1, Th30 ; v1 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) = w1 by A1, A2, A4, Th36; then A7: w1 + w1 = ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) + (v1 + (v1 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)))) by RLVECT_1:def_3 .= ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) + ((v1 + v1) + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9))) by RLVECT_1:def_3 .= (v1 + v1) + (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9))) by RLVECT_1:def_3 .= (v1 + v1) + (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) + (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)) by RLVECT_1:def_6 ; ( u1 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) = t1 & u2 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) = t2 ) by A1, A2, A3, Th36; then A8: t1 + t2 = ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) + (u1 + (u2 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)))) by RLVECT_1:def_3 .= ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) + ((u1 + u2) + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9))) by RLVECT_1:def_3 .= (u1 + u2) + (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9))) by RLVECT_1:def_3 .= (u1 + u2) + (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) + (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)) by RLVECT_1:def_6 .= (v1 + v1) + (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) + (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)) by A5, Def2 ; set vv = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) + ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))); (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) + (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) = (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) + ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2))))) * ((PProJ (w,y,(u - u9),(u - u9))) ") .= (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) + ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1))))) * ((PProJ (w,y,(u - u9),(u - u9))) ") by A6 .= (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) + (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) ; hence w1 = t1 # t2 by A8, A7, Def2; ::_thesis: verum end; theorem Th39: :: GEOMTRAP:39 for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for u, u9, u1, u2, t1, t2 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y holds u,u9,u1 # u2,t1 # t2 are_DTr_wrt w,y proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V st Gen w,y holds for u, u9, u1, u2, t1, t2 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y holds u,u9,u1 # u2,t1 # t2 are_DTr_wrt w,y let w, y be VECTOR of V; ::_thesis: ( Gen w,y implies for u, u9, u1, u2, t1, t2 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y holds u,u9,u1 # u2,t1 # t2 are_DTr_wrt w,y ) assume A1: Gen w,y ; ::_thesis: for u, u9, u1, u2, t1, t2 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y holds u,u9,u1 # u2,t1 # t2 are_DTr_wrt w,y let u, u9, u1, u2, t1, t2 be VECTOR of V; ::_thesis: ( u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y implies u,u9,u1 # u2,t1 # t2 are_DTr_wrt w,y ) assume that A2: u <> u9 and A3: u,u9,u1,t1 are_DTr_wrt w,y and A4: u,u9,u2,t2 are_DTr_wrt w,y ; ::_thesis: u,u9,u1 # u2,t1 # t2 are_DTr_wrt w,y u1,t1,u2,t2 are_DTr_wrt w,y by A1, A2, A3, A4, Th19; then A5: u1,t1,u1 # u2,t1 # t2 are_DTr_wrt w,y by A1, Th26; u2,t2,u1,t1 are_DTr_wrt w,y by A1, A2, A3, A4, Th19; then A6: u2,t2,u2 # u1,t2 # t1 are_DTr_wrt w,y by A1, Th26; percases ( u1 <> t1 or u2 <> t2 or ( u1 = t1 & u2 = t2 ) ) ; supposeA7: u1 <> t1 ; ::_thesis: u,u9,u1 # u2,t1 # t2 are_DTr_wrt w,y u1,t1,u,u9 are_DTr_wrt w,y by A3, Th21; hence u,u9,u1 # u2,t1 # t2 are_DTr_wrt w,y by A1, A5, A7, Th19; ::_thesis: verum end; supposeA8: u2 <> t2 ; ::_thesis: u,u9,u1 # u2,t1 # t2 are_DTr_wrt w,y u2,t2,u,u9 are_DTr_wrt w,y by A4, Th21; hence u,u9,u1 # u2,t1 # t2 are_DTr_wrt w,y by A1, A6, A8, Th19; ::_thesis: verum end; supposethat A9: u1 = t1 and A10: u2 = t2 ; ::_thesis: u,u9,u1 # u2,t1 # t2 are_DTr_wrt w,y u # u9,(u1 # u2) # (t1 # t2),u1 # u2,t1 # t2 are_Ort_wrt w,y by A1, A9, A10, Lm6; then A11: u1 # u2,t1 # t2,u # u9,(u1 # u2) # (t1 # t2) are_Ort_wrt w,y by Lm5; A12: u,u9,u # u9,u1 # u2 are_Ort_wrt w,y proof set uu9 = u # u9; A13: 2 * (u1 # u2) = (1 + 1) * (u1 # u2) .= (1 * (u1 # u2)) + (1 * (u1 # u2)) by RLVECT_1:def_6 .= (u1 # u2) + (1 * (u1 # u2)) by RLVECT_1:def_8 .= (u1 # u2) + (u1 # u2) by RLVECT_1:def_8 .= u1 + u2 by Def2 ; A14: - (2 * (u # u9)) = - ((1 + 1) * (u # u9)) .= - ((1 * (u # u9)) + (1 * (u # u9))) by RLVECT_1:def_6 .= - ((u # u9) + (1 * (u # u9))) by RLVECT_1:def_8 .= - ((u # u9) + (u # u9)) by RLVECT_1:def_8 .= (- (u # u9)) + (- (u # u9)) by Lm19 ; u,u9,u # u9,u2 # t2 are_Ort_wrt w,y by A4, Def3; then u9 - u,u2 - (u # u9) are_Ort_wrt w,y by A10, ANALMETR:def_3; then A15: u9 - u,(2 ") * (u2 - (u # u9)) are_Ort_wrt w,y by ANALMETR:7; u,u9,u # u9,u1 # t1 are_Ort_wrt w,y by A3, Def3; then u9 - u,u1 - (u # u9) are_Ort_wrt w,y by A9, ANALMETR:def_3; then A16: u9 - u,(2 ") * (u1 - (u # u9)) are_Ort_wrt w,y by ANALMETR:7; (u1 # u2) - (u # u9) = ((2 ") * 2) * ((u1 # u2) - (u # u9)) by RLVECT_1:def_8 .= (2 ") * (2 * ((u1 # u2) - (u # u9))) by RLVECT_1:def_7 .= (2 ") * ((u1 + u2) - (2 * (u # u9))) by A13, RLVECT_1:34 .= (2 ") * (((u1 + u2) + (- (u # u9))) + (- (u # u9))) by A14, RLVECT_1:def_3 .= (2 ") * (((u1 - (u # u9)) + u2) + (- (u # u9))) by RLVECT_1:def_3 .= (2 ") * ((u1 - (u # u9)) + (u2 - (u # u9))) by RLVECT_1:def_3 .= ((2 ") * (u1 - (u # u9))) + ((2 ") * (u2 - (u # u9))) by RLVECT_1:def_5 ; then u9 - u,(u1 # u2) - (u # u9) are_Ort_wrt w,y by A1, A16, A15, ANALMETR:10; hence u,u9,u # u9,u1 # u2 are_Ort_wrt w,y by ANALMETR:def_3; ::_thesis: verum end; u,u9 // u1 # u2,t1 # t2 by A9, A10, ANALOAF:9; hence u,u9,u1 # u2,t1 # t2 are_DTr_wrt w,y by A9, A10, A11, A12, Def3; ::_thesis: verum end; end; end; theorem Th40: :: GEOMTRAP:40 for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for u, u9, u1, u2, v1, v2, t1, t2, w1, w2 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2,v1,v2 are_Ort_wrt w,y holds t1,t2,w1,w2 are_Ort_wrt w,y proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V st Gen w,y holds for u, u9, u1, u2, v1, v2, t1, t2, w1, w2 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2,v1,v2 are_Ort_wrt w,y holds t1,t2,w1,w2 are_Ort_wrt w,y let w, y be VECTOR of V; ::_thesis: ( Gen w,y implies for u, u9, u1, u2, v1, v2, t1, t2, w1, w2 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2,v1,v2 are_Ort_wrt w,y holds t1,t2,w1,w2 are_Ort_wrt w,y ) assume A1: Gen w,y ; ::_thesis: for u, u9, u1, u2, v1, v2, t1, t2, w1, w2 being VECTOR of V st u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2,v1,v2 are_Ort_wrt w,y holds t1,t2,w1,w2 are_Ort_wrt w,y let u, u9, u1, u2, v1, v2, t1, t2, w1, w2 be VECTOR of V; ::_thesis: ( u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2,v1,v2 are_Ort_wrt w,y implies t1,t2,w1,w2 are_Ort_wrt w,y ) assume that A2: u <> u9 and A3: ( u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y ) and A4: ( u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y ) and A5: u1,u2,v1,v2 are_Ort_wrt w,y ; ::_thesis: t1,t2,w1,w2 are_Ort_wrt w,y set A1 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "); set A2 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) "); set A3 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "); set A4 = ((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) "); ( u1 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) = t1 & u2 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) = t2 ) by A1, A2, A3, Th36; then A6: t2 - t1 = (u2 - u1) + (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)) by A1, Lm21; A7: ( t1 = u1 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) & t2 = u2 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) ) by A1, A2, A3, Th36; A8: ( v1 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) = w1 & v2 + ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (u - u9)) = w2 ) by A1, A2, A4, Th36; then w2 - w1 = (v2 - v1) + (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)) by A1, Lm21; then A9: PProJ (w,y,(t2 - t1),(w2 - w1)) = (PProJ (w,y,(u2 - u1),((v2 - v1) + (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9))))) + (PProJ (w,y,(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)),((v2 - v1) + (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9))))) by A1, A6, Th30 .= ((PProJ (w,y,(u2 - u1),(v2 - v1))) + (PProJ (w,y,(u2 - u1),(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9))))) + (PProJ (w,y,(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)),((v2 - v1) + (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9))))) by A1, Th30 .= (0 + (PProJ (w,y,(u2 - u1),(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9))))) + (PProJ (w,y,(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)),((v2 - v1) + (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9))))) by A1, A5, Th33 .= (PProJ (w,y,(u2 - u1),(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)))) + ((PProJ (w,y,(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)),(v2 - v1))) + (PProJ (w,y,(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)),(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9))))) by A1, Th30 ; set aa = ((PProJ (w,y,(u - u9),(u - u9))) ") * ((PProJ (w,y,(u - u9),(u2 - u1))) * (PProJ (w,y,(u - u9),(v2 - v1)))); A10: PProJ (w,y,(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)),(v2 - v1)) = PProJ (w,y,(v2 - v1),(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9))) .= (PProJ (w,y,(v2 - v1),(u - u9))) * ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) by A1, Th31 .= (PProJ (w,y,(v2 - v1),(u - u9))) * (((- 2) * (PProJ (w,y,(u - u9),(u2 - u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) by A1, A7, Lm21 .= (- 2) * (((PProJ (w,y,(u - u9),(u - u9))) ") * ((PProJ (w,y,(u - u9),(u2 - u1))) * (PProJ (w,y,(u - u9),(v2 - v1))))) ; A11: PProJ (w,y,(u - u9),(u - u9)) <> 0 by A1, A2, Th35; A12: PProJ (w,y,(u2 - u1),(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9))) = (PProJ (w,y,(u2 - u1),(u - u9))) * ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) by A1, Th31 .= (PProJ (w,y,(u2 - u1),(u - u9))) * (((- 2) * (PProJ (w,y,(u - u9),(v2 - v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) by A1, A8, Lm21 .= (- 2) * (((PProJ (w,y,(u - u9),(u - u9))) ") * ((PProJ (w,y,(u - u9),(u2 - u1))) * (PProJ (w,y,(u - u9),(v2 - v1))))) ; PProJ (w,y,(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)),(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9))) = ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (PProJ (w,y,(u - u9),(((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (u - u9)))) by A1, Th31 .= ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (PProJ (w,y,(u - u9),(u - u9)))) by A1, Th31 .= ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * ((((- 2) * (PProJ (w,y,(u - u9),(v2 - v1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) * (PProJ (w,y,(u - u9),(u - u9)))) by A1, A8, Lm21 .= ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (((- 2) * (PProJ (w,y,(u - u9),(v2 - v1)))) * (((PProJ (w,y,(u - u9),(u - u9))) ") * (PProJ (w,y,(u - u9),(u - u9))))) .= ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (((- 2) * (PProJ (w,y,(u - u9),(v2 - v1)))) * 1) by A11, XCMPLX_0:def_7 .= ((- 2) * ((((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u2)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")) - (((PProJ (w,y,(u - u9),(u + u9))) - (2 * (PProJ (w,y,(u - u9),u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) ")))) * (PProJ (w,y,(u - u9),(v2 - v1))) .= ((- 2) * (((- 2) * (PProJ (w,y,(u - u9),(u2 - u1)))) * ((PProJ (w,y,(u - u9),(u - u9))) "))) * (PProJ (w,y,(u - u9),(v2 - v1))) by A1, A7, Lm21 .= 4 * (((PProJ (w,y,(u - u9),(u - u9))) ") * ((PProJ (w,y,(u - u9),(u2 - u1))) * (PProJ (w,y,(u - u9),(v2 - v1))))) ; hence t1,t2,w1,w2 are_Ort_wrt w,y by A1, A9, A12, A10, Th33; ::_thesis: verum end; theorem Th41: :: GEOMTRAP:41 for V being RealLinearSpace for w, y, u, u9, u1, u2, v1, v2, t1, t2, w1, w2 being VECTOR of V st Gen w,y & u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2,v1,v2 are_DTr_wrt w,y holds t1,t2,w1,w2 are_DTr_wrt w,y proof let V be RealLinearSpace; ::_thesis: for w, y, u, u9, u1, u2, v1, v2, t1, t2, w1, w2 being VECTOR of V st Gen w,y & u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2,v1,v2 are_DTr_wrt w,y holds t1,t2,w1,w2 are_DTr_wrt w,y let w, y be VECTOR of V; ::_thesis: for u, u9, u1, u2, v1, v2, t1, t2, w1, w2 being VECTOR of V st Gen w,y & u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2,v1,v2 are_DTr_wrt w,y holds t1,t2,w1,w2 are_DTr_wrt w,y let u, u9, u1, u2, v1, v2, t1, t2, w1, w2 be VECTOR of V; ::_thesis: ( Gen w,y & u <> u9 & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2,v1,v2 are_DTr_wrt w,y implies t1,t2,w1,w2 are_DTr_wrt w,y ) assume that A1: ( Gen w,y & u <> u9 ) and A2: ( u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y ) and A3: ( u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y ) and A4: u1,u2,v1,v2 are_DTr_wrt w,y ; ::_thesis: t1,t2,w1,w2 are_DTr_wrt w,y set uu = u1 # u2; set vv = v1 # v2; A5: ( u,u9,u1 # u2,t1 # t2 are_DTr_wrt w,y & u,u9,v1 # v2,w1 # w2 are_DTr_wrt w,y ) by A1, A2, A3, Th39; v1,v2,u1 # u2,v1 # v2 are_Ort_wrt w,y by A4, Def3; then A6: w1,w2,t1 # t2,w1 # w2 are_Ort_wrt w,y by A1, A3, A5, Th40; u1,u2,u1 # u2,v1 # v2 are_Ort_wrt w,y by A4, Def3; then A7: t1,t2,t1 # t2,w1 # w2 are_Ort_wrt w,y by A1, A2, A5, Th40; u1,u2 // v1,v2 by A4, Def3; then t1,t2 // w1,w2 by A1, A2, A3, Th37; hence t1,t2,w1,w2 are_DTr_wrt w,y by A7, A6, Def3; ::_thesis: verum end; definition let V be RealLinearSpace; let w, y be VECTOR of V; func DTrapezium (V,w,y) -> Relation of [: the carrier of V, the carrier of V:] means :Def7: :: GEOMTRAP:def 7 for x, z being set holds ( [x,z] in it iff ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ); existence ex b1 being Relation of [: the carrier of V, the carrier of V:] st for x, z being set holds ( [x,z] in b1 iff ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ) proof defpred S1[ set , set ] means ex u, u1, v, v1 being VECTOR of V st ( $1 = [u,u1] & $2 = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ); set VV = [: the carrier of V, the carrier of V:]; consider P being Relation of [: the carrier of V, the carrier of V:],[: the carrier of V, the carrier of V:] such that A1: for x, z being set holds ( [x,z] in P iff ( x in [: the carrier of V, the carrier of V:] & z in [: the carrier of V, the carrier of V:] & S1[x,z] ) ) from RELSET_1:sch_1(); take P ; ::_thesis: for x, z being set holds ( [x,z] in P iff ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ) let x, z be set ; ::_thesis: ( [x,z] in P iff ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ) thus ( [x,z] in P implies ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ) by A1; ::_thesis: ( ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) implies [x,z] in P ) assume ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ; ::_thesis: [x,z] in P hence [x,z] in P by A1; ::_thesis: verum end; uniqueness for b1, b2 being Relation of [: the carrier of V, the carrier of V:] st ( for x, z being set holds ( [x,z] in b1 iff ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ) ) & ( for x, z being set holds ( [x,z] in b2 iff ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ) ) holds b1 = b2 proof let P, Q be Relation of [: the carrier of V, the carrier of V:]; ::_thesis: ( ( for x, z being set holds ( [x,z] in P iff ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ) ) & ( for x, z being set holds ( [x,z] in Q iff ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ) ) implies P = Q ) assume that A2: for x, z being set holds ( [x,z] in P iff ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ) and A3: for x, z being set holds ( [x,z] in Q iff ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ) ; ::_thesis: P = Q for x, z being set holds ( [x,z] in P iff [x,z] in Q ) proof let x, z be set ; ::_thesis: ( [x,z] in P iff [x,z] in Q ) ( [x,z] in P iff ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ) by A2; hence ( [x,z] in P iff [x,z] in Q ) by A3; ::_thesis: verum end; hence P = Q by RELAT_1:def_2; ::_thesis: verum end; end; :: deftheorem Def7 defines DTrapezium GEOMTRAP:def_7_:_ for V being RealLinearSpace for w, y being VECTOR of V for b4 being Relation of [: the carrier of V, the carrier of V:] holds ( b4 = DTrapezium (V,w,y) iff for x, z being set holds ( [x,z] in b4 iff ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ) ); theorem Th42: :: GEOMTRAP:42 for V being RealLinearSpace for u, v, u1, v1, w, y being VECTOR of V holds ( [[u,v],[u1,v1]] in DTrapezium (V,w,y) iff u,v,u1,v1 are_DTr_wrt w,y ) proof let V be RealLinearSpace; ::_thesis: for u, v, u1, v1, w, y being VECTOR of V holds ( [[u,v],[u1,v1]] in DTrapezium (V,w,y) iff u,v,u1,v1 are_DTr_wrt w,y ) let u, v, u1, v1, w, y be VECTOR of V; ::_thesis: ( [[u,v],[u1,v1]] in DTrapezium (V,w,y) iff u,v,u1,v1 are_DTr_wrt w,y ) now__::_thesis:_(_[[u,v],[u1,v1]]_in_DTrapezium_(V,w,y)_implies_u,v,u1,v1_are_DTr_wrt_w,y_) assume [[u,v],[u1,v1]] in DTrapezium (V,w,y) ; ::_thesis: u,v,u1,v1 are_DTr_wrt w,y then consider u9, v9, u19, v19 being VECTOR of V such that A1: [u,v] = [u9,v9] and A2: [u1,v1] = [u19,v19] and A3: u9,v9,u19,v19 are_DTr_wrt w,y by Def7; A4: u1 = u19 by A2, XTUPLE_0:1; ( u = u9 & v = v9 ) by A1, XTUPLE_0:1; hence u,v,u1,v1 are_DTr_wrt w,y by A2, A3, A4, XTUPLE_0:1; ::_thesis: verum end; hence ( [[u,v],[u1,v1]] in DTrapezium (V,w,y) iff u,v,u1,v1 are_DTr_wrt w,y ) by Def7; ::_thesis: verum end; definition let V be RealLinearSpace; func MidPoint V -> BinOp of the carrier of V means :Def8: :: GEOMTRAP:def 8 for u, v being VECTOR of V holds it . (u,v) = u # v; existence ex b1 being BinOp of the carrier of V st for u, v being VECTOR of V holds b1 . (u,v) = u # v proof deffunc H1( VECTOR of V, VECTOR of V) -> VECTOR of V = $1 # $2; ex B being BinOp of the carrier of V st for u, v being VECTOR of V holds B . (u,v) = H1(u,v) from BINOP_1:sch_4(); hence ex b1 being BinOp of the carrier of V st for u, v being VECTOR of V holds b1 . (u,v) = u # v ; ::_thesis: verum end; uniqueness for b1, b2 being BinOp of the carrier of V st ( for u, v being VECTOR of V holds b1 . (u,v) = u # v ) & ( for u, v being VECTOR of V holds b2 . (u,v) = u # v ) holds b1 = b2 proof deffunc H1( VECTOR of V, VECTOR of V) -> VECTOR of V = $1 # $2; for o1, o2 being BinOp of the carrier of V st ( for a, b being Element of V holds o1 . (a,b) = H1(a,b) ) & ( for a, b being Element of V holds o2 . (a,b) = H1(a,b) ) holds o1 = o2 from BINOP_2:sch_2(); hence for b1, b2 being BinOp of the carrier of V st ( for u, v being VECTOR of V holds b1 . (u,v) = u # v ) & ( for u, v being VECTOR of V holds b2 . (u,v) = u # v ) holds b1 = b2 ; ::_thesis: verum end; end; :: deftheorem Def8 defines MidPoint GEOMTRAP:def_8_:_ for V being RealLinearSpace for b2 being BinOp of the carrier of V holds ( b2 = MidPoint V iff for u, v being VECTOR of V holds b2 . (u,v) = u # v ); definition attrc1 is strict ; struct AfMidStruct -> AffinStruct , MidStr ; aggrAfMidStruct(# carrier, MIDPOINT, CONGR #) -> AfMidStruct ; end; registration cluster non empty strict for AfMidStruct ; existence ex b1 being AfMidStruct st ( not b1 is empty & b1 is strict ) proof set D = the non empty set ; set m = the BinOp of the non empty set ; set c = the Relation of [: the non empty set , the non empty set :]; take AfMidStruct(# the non empty set , the BinOp of the non empty set , the Relation of [: the non empty set , the non empty set :] #) ; ::_thesis: ( not AfMidStruct(# the non empty set , the BinOp of the non empty set , the Relation of [: the non empty set , the non empty set :] #) is empty & AfMidStruct(# the non empty set , the BinOp of the non empty set , the Relation of [: the non empty set , the non empty set :] #) is strict ) thus not the carrier of AfMidStruct(# the non empty set , the BinOp of the non empty set , the Relation of [: the non empty set , the non empty set :] #) is empty ; :: according to STRUCT_0:def_1 ::_thesis: AfMidStruct(# the non empty set , the BinOp of the non empty set , the Relation of [: the non empty set , the non empty set :] #) is strict thus AfMidStruct(# the non empty set , the BinOp of the non empty set , the Relation of [: the non empty set , the non empty set :] #) is strict ; ::_thesis: verum end; end; definition let V be RealLinearSpace; let w, y be VECTOR of V; func DTrSpace (V,w,y) -> strict AfMidStruct equals :: GEOMTRAP:def 9 AfMidStruct(# the carrier of V,(MidPoint V),(DTrapezium (V,w,y)) #); correctness coherence AfMidStruct(# the carrier of V,(MidPoint V),(DTrapezium (V,w,y)) #) is strict AfMidStruct ; ; end; :: deftheorem defines DTrSpace GEOMTRAP:def_9_:_ for V being RealLinearSpace for w, y being VECTOR of V holds DTrSpace (V,w,y) = AfMidStruct(# the carrier of V,(MidPoint V),(DTrapezium (V,w,y)) #); registration let V be RealLinearSpace; let w, y be VECTOR of V; cluster DTrSpace (V,w,y) -> non empty strict ; coherence not DTrSpace (V,w,y) is empty ; end; definition let AMS be AfMidStruct ; func Af AMS -> strict AffinStruct equals :: GEOMTRAP:def 10 AffinStruct(# the carrier of AMS, the CONGR of AMS #); correctness coherence AffinStruct(# the carrier of AMS, the CONGR of AMS #) is strict AffinStruct ; ; end; :: deftheorem defines Af GEOMTRAP:def_10_:_ for AMS being AfMidStruct holds Af AMS = AffinStruct(# the carrier of AMS, the CONGR of AMS #); registration let AMS be non empty AfMidStruct ; cluster Af AMS -> non empty strict ; coherence not Af AMS is empty ; end; definition let AMS be non empty AfMidStruct ; let a, b be Element of AMS; funca # b -> Element of AMS equals :: GEOMTRAP:def 11 the MIDPOINT of AMS . (a,b); correctness coherence the MIDPOINT of AMS . (a,b) is Element of AMS; ; end; :: deftheorem defines # GEOMTRAP:def_11_:_ for AMS being non empty AfMidStruct for a, b being Element of AMS holds a # b = the MIDPOINT of AMS . (a,b); theorem :: GEOMTRAP:43 for V being RealLinearSpace for w, y being VECTOR of V for x being set holds ( x is Element of the carrier of (DTrSpace (V,w,y)) iff x is VECTOR of V ) ; theorem Th44: :: GEOMTRAP:44 for V being RealLinearSpace for u, v, u1, v1, w, y being VECTOR of V for a, b, a1, b1 being Element of (DTrSpace (V,w,y)) st u = a & v = b & u1 = a1 & v1 = b1 holds ( a,b // a1,b1 iff u,v,u1,v1 are_DTr_wrt w,y ) proof let V be RealLinearSpace; ::_thesis: for u, v, u1, v1, w, y being VECTOR of V for a, b, a1, b1 being Element of (DTrSpace (V,w,y)) st u = a & v = b & u1 = a1 & v1 = b1 holds ( a,b // a1,b1 iff u,v,u1,v1 are_DTr_wrt w,y ) let u, v, u1, v1, w, y be VECTOR of V; ::_thesis: for a, b, a1, b1 being Element of (DTrSpace (V,w,y)) st u = a & v = b & u1 = a1 & v1 = b1 holds ( a,b // a1,b1 iff u,v,u1,v1 are_DTr_wrt w,y ) let a, b, a1, b1 be Element of (DTrSpace (V,w,y)); ::_thesis: ( u = a & v = b & u1 = a1 & v1 = b1 implies ( a,b // a1,b1 iff u,v,u1,v1 are_DTr_wrt w,y ) ) assume A1: ( u = a & v = b & u1 = a1 & v1 = b1 ) ; ::_thesis: ( a,b // a1,b1 iff u,v,u1,v1 are_DTr_wrt w,y ) hereby ::_thesis: ( u,v,u1,v1 are_DTr_wrt w,y implies a,b // a1,b1 ) assume a,b // a1,b1 ; ::_thesis: u,v,u1,v1 are_DTr_wrt w,y then [[a,b],[a1,b1]] in DTrapezium (V,w,y) by ANALOAF:def_2; hence u,v,u1,v1 are_DTr_wrt w,y by A1, Th42; ::_thesis: verum end; assume u,v,u1,v1 are_DTr_wrt w,y ; ::_thesis: a,b // a1,b1 then [[a,b],[a1,b1]] in DTrapezium (V,w,y) by A1, Th42; hence a,b // a1,b1 by ANALOAF:def_2; ::_thesis: verum end; theorem :: GEOMTRAP:45 for V being RealLinearSpace for w, y, u, v being VECTOR of V for a, b being Element of (DTrSpace (V,w,y)) st Gen w,y & u = a & v = b holds u # v = a # b by Def8; Lm22: for V being RealLinearSpace for w, y being VECTOR of V for a, b, c being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // b,c holds ( a = b & b = c ) proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V for a, b, c being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // b,c holds ( a = b & b = c ) let w, y be VECTOR of V; ::_thesis: for a, b, c being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // b,c holds ( a = b & b = c ) let a, b, c be Element of (DTrSpace (V,w,y)); ::_thesis: ( Gen w,y & a,b // b,c implies ( a = b & b = c ) ) assume that A1: Gen w,y and A2: a,b // b,c ; ::_thesis: ( a = b & b = c ) reconsider u = a, v = b, u1 = c as VECTOR of V ; u,v,v,u1 are_DTr_wrt w,y by A2, Th44; hence ( a = b & b = c ) by A1, Th18; ::_thesis: verum end; Lm23: for V being RealLinearSpace for w, y being VECTOR of V for a, b, a1, b1, c1, d1 being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // a1,b1 & a,b // c1,d1 & a <> b holds a1,b1 // c1,d1 proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V for a, b, a1, b1, c1, d1 being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // a1,b1 & a,b // c1,d1 & a <> b holds a1,b1 // c1,d1 let w, y be VECTOR of V; ::_thesis: for a, b, a1, b1, c1, d1 being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // a1,b1 & a,b // c1,d1 & a <> b holds a1,b1 // c1,d1 let a, b, a1, b1, c1, d1 be Element of (DTrSpace (V,w,y)); ::_thesis: ( Gen w,y & a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) assume that A1: Gen w,y and A2: ( a,b // a1,b1 & a,b // c1,d1 ) and A3: a <> b ; ::_thesis: a1,b1 // c1,d1 reconsider u = a, v = b, u1 = a1, v1 = b1, u2 = c1, v2 = d1 as VECTOR of V ; ( u,v,u1,v1 are_DTr_wrt w,y & u,v,u2,v2 are_DTr_wrt w,y ) by A2, Th44; then u1,v1,u2,v2 are_DTr_wrt w,y by A1, A3, Th19; hence a1,b1 // c1,d1 by Th44; ::_thesis: verum end; Lm24: for V being RealLinearSpace for w, y being VECTOR of V for a, b, c, d being Element of (DTrSpace (V,w,y)) st a,b // c,d holds ( c,d // a,b & b,a // d,c ) proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V for a, b, c, d being Element of (DTrSpace (V,w,y)) st a,b // c,d holds ( c,d // a,b & b,a // d,c ) let w, y be VECTOR of V; ::_thesis: for a, b, c, d being Element of (DTrSpace (V,w,y)) st a,b // c,d holds ( c,d // a,b & b,a // d,c ) let a, b, c, d be Element of (DTrSpace (V,w,y)); ::_thesis: ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) reconsider u = a, v = b, u1 = c, v1 = d as VECTOR of V ; assume a,b // c,d ; ::_thesis: ( c,d // a,b & b,a // d,c ) then u,v,u1,v1 are_DTr_wrt w,y by Th44; then ( v,u,v1,u1 are_DTr_wrt w,y & u1,v1,u,v are_DTr_wrt w,y ) by Th21, Th22; hence ( c,d // a,b & b,a // d,c ) by Th44; ::_thesis: verum end; Lm25: for V being RealLinearSpace for w, y being VECTOR of V for a, b, c being Element of (DTrSpace (V,w,y)) st Gen w,y holds ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V for a, b, c being Element of (DTrSpace (V,w,y)) st Gen w,y holds ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) let w, y be VECTOR of V; ::_thesis: for a, b, c being Element of (DTrSpace (V,w,y)) st Gen w,y holds ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) let a, b, c be Element of (DTrSpace (V,w,y)); ::_thesis: ( Gen w,y implies ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) ) reconsider u = a, v = b, u1 = c as VECTOR of V ; assume Gen w,y ; ::_thesis: ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) then consider t being VECTOR of V such that A1: ( u,v,u1,t are_DTr_wrt w,y or u,v,t,u1 are_DTr_wrt w,y ) by Th20; reconsider d = t as Element of (DTrSpace (V,w,y)) ; ( a,b // c,d or a,b // d,c ) by A1, Th44; hence ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) ; ::_thesis: verum end; Lm26: for V being RealLinearSpace for w, y being VECTOR of V for a, b, c, d1, d2 being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // c,d1 & a,b // c,d2 & not a = b holds d1 = d2 proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V for a, b, c, d1, d2 being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // c,d1 & a,b // c,d2 & not a = b holds d1 = d2 let w, y be VECTOR of V; ::_thesis: for a, b, c, d1, d2 being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // c,d1 & a,b // c,d2 & not a = b holds d1 = d2 let a, b, c, d1, d2 be Element of (DTrSpace (V,w,y)); ::_thesis: ( Gen w,y & a,b // c,d1 & a,b // c,d2 & not a = b implies d1 = d2 ) assume that A1: Gen w,y and A2: ( a,b // c,d1 & a,b // c,d2 ) ; ::_thesis: ( a = b or d1 = d2 ) reconsider u = a, v = b, u1 = c, v1 = d1, v2 = d2 as VECTOR of V ; ( u,v,u1,v1 are_DTr_wrt w,y & u,v,u1,v2 are_DTr_wrt w,y ) by A2, Th44; hence ( a = b or d1 = d2 ) by A1, Th24; ::_thesis: verum end; Lm27: for V being RealLinearSpace for w, y being VECTOR of V for a, b being Element of (DTrSpace (V,w,y)) holds a # b = b # a proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V for a, b being Element of (DTrSpace (V,w,y)) holds a # b = b # a let w, y be VECTOR of V; ::_thesis: for a, b being Element of (DTrSpace (V,w,y)) holds a # b = b # a let a, b be Element of (DTrSpace (V,w,y)); ::_thesis: a # b = b # a reconsider u = a, v = b as VECTOR of V ; thus a # b = u # v by Def8 .= b # a by Def8 ; ::_thesis: verum end; Lm28: for V being RealLinearSpace for w, y being VECTOR of V for a being Element of (DTrSpace (V,w,y)) holds a # a = a proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V for a being Element of (DTrSpace (V,w,y)) holds a # a = a let w, y be VECTOR of V; ::_thesis: for a being Element of (DTrSpace (V,w,y)) holds a # a = a let a be Element of (DTrSpace (V,w,y)); ::_thesis: a # a = a reconsider u = a as VECTOR of V ; u # u = u ; hence a # a = a by Def8; ::_thesis: verum end; Lm29: for V being RealLinearSpace for w, y being VECTOR of V for a, b, c, d being Element of (DTrSpace (V,w,y)) holds (a # b) # (c # d) = (a # c) # (b # d) proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V for a, b, c, d being Element of (DTrSpace (V,w,y)) holds (a # b) # (c # d) = (a # c) # (b # d) let w, y be VECTOR of V; ::_thesis: for a, b, c, d being Element of (DTrSpace (V,w,y)) holds (a # b) # (c # d) = (a # c) # (b # d) let a, b, c, d be Element of (DTrSpace (V,w,y)); ::_thesis: (a # b) # (c # d) = (a # c) # (b # d) reconsider u = a, u1 = b, v = c, v1 = d as VECTOR of V ; set ab = a # b; set cd = c # d; set ac = a # c; set bd = b # d; set uu1 = u # u1; set vv1 = v # v1; set uv = u # v; set u1v1 = u1 # v1; A1: ( a # c = u # v & b # d = u1 # v1 ) by Def8; ( a # b = u # u1 & c # d = v # v1 ) by Def8; hence (a # b) # (c # d) = (u # u1) # (v # v1) by Def8 .= (u # v) # (u1 # v1) by Th6 .= (a # c) # (b # d) by A1, Def8 ; ::_thesis: verum end; Lm30: for V being RealLinearSpace for y, w being VECTOR of V for a, b being Element of (DTrSpace (V,w,y)) ex p being Element of (DTrSpace (V,w,y)) st p # a = b proof let V be RealLinearSpace; ::_thesis: for y, w being VECTOR of V for a, b being Element of (DTrSpace (V,w,y)) ex p being Element of (DTrSpace (V,w,y)) st p # a = b let y, w be VECTOR of V; ::_thesis: for a, b being Element of (DTrSpace (V,w,y)) ex p being Element of (DTrSpace (V,w,y)) st p # a = b let a, b be Element of (DTrSpace (V,w,y)); ::_thesis: ex p being Element of (DTrSpace (V,w,y)) st p # a = b reconsider u = a, v = b as VECTOR of V ; consider u1 being VECTOR of V such that A1: u # u1 = v by Th5; reconsider p = u1 as Element of (DTrSpace (V,w,y)) ; p # a = u # u1 by Def8; hence ex p being Element of (DTrSpace (V,w,y)) st p # a = b by A1; ::_thesis: verum end; Lm31: for V being RealLinearSpace for w, y being VECTOR of V for a, a1, a2 being Element of (DTrSpace (V,w,y)) st a # a1 = a # a2 holds a1 = a2 proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V for a, a1, a2 being Element of (DTrSpace (V,w,y)) st a # a1 = a # a2 holds a1 = a2 let w, y be VECTOR of V; ::_thesis: for a, a1, a2 being Element of (DTrSpace (V,w,y)) st a # a1 = a # a2 holds a1 = a2 let a, a1, a2 be Element of (DTrSpace (V,w,y)); ::_thesis: ( a # a1 = a # a2 implies a1 = a2 ) assume A1: a # a1 = a # a2 ; ::_thesis: a1 = a2 reconsider u = a, u1 = a1, u2 = a2 as VECTOR of V ; ( u # u1 = a # a1 & u # u2 = a # a2 ) by Def8; hence a1 = a2 by A1, Th7; ::_thesis: verum end; Lm32: for V being RealLinearSpace for w, y being VECTOR of V for a, b, c, d being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // c,d holds a,b // a # c,b # d proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V for a, b, c, d being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // c,d holds a,b // a # c,b # d let w, y be VECTOR of V; ::_thesis: for a, b, c, d being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // c,d holds a,b // a # c,b # d let a, b, c, d be Element of (DTrSpace (V,w,y)); ::_thesis: ( Gen w,y & a,b // c,d implies a,b // a # c,b # d ) assume that A1: Gen w,y and A2: a,b // c,d ; ::_thesis: a,b // a # c,b # d reconsider u = a, v = b, u1 = c, v1 = d as VECTOR of V ; u,v,u1,v1 are_DTr_wrt w,y by A2, Th44; then A3: u,v,u # u1,v # v1 are_DTr_wrt w,y by A1, Th26; ( a # c = u # u1 & b # d = v # v1 ) by Def8; hence a,b // a # c,b # d by A3, Th44; ::_thesis: verum end; Lm33: for V being RealLinearSpace for w, y being VECTOR of V for a, b, c, d being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // c,d & not a,b // a # d,b # c holds a,b // b # c,a # d proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V for a, b, c, d being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // c,d & not a,b // a # d,b # c holds a,b // b # c,a # d let w, y be VECTOR of V; ::_thesis: for a, b, c, d being Element of (DTrSpace (V,w,y)) st Gen w,y & a,b // c,d & not a,b // a # d,b # c holds a,b // b # c,a # d let a, b, c, d be Element of (DTrSpace (V,w,y)); ::_thesis: ( Gen w,y & a,b // c,d & not a,b // a # d,b # c implies a,b // b # c,a # d ) assume that A1: Gen w,y and A2: a,b // c,d ; ::_thesis: ( a,b // a # d,b # c or a,b // b # c,a # d ) reconsider u = a, v = b, u1 = c, v1 = d as VECTOR of V ; u,v,u1,v1 are_DTr_wrt w,y by A2, Th44; then A3: ( u,v,u # v1,v # u1 are_DTr_wrt w,y or u,v,v # u1,u # v1 are_DTr_wrt w,y ) by A1, Th27; ( a # d = u # v1 & b # c = v # u1 ) by Def8; hence ( a,b // a # d,b # c or a,b // b # c,a # d ) by A3, Th44; ::_thesis: verum end; Lm34: for V being RealLinearSpace for w, y being VECTOR of V for a, b, c, d, a1, p, b1, c1, d1 being Element of (DTrSpace (V,w,y)) st a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p holds a1,b1 // c1,d1 proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V for a, b, c, d, a1, p, b1, c1, d1 being Element of (DTrSpace (V,w,y)) st a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p holds a1,b1 // c1,d1 let w, y be VECTOR of V; ::_thesis: for a, b, c, d, a1, p, b1, c1, d1 being Element of (DTrSpace (V,w,y)) st a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p holds a1,b1 // c1,d1 let a, b, c, d, a1, p, b1, c1, d1 be Element of (DTrSpace (V,w,y)); ::_thesis: ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) assume that A1: ( a,b // c,d & a # a1 = p ) and A2: ( b # b1 = p & c # c1 = p ) and A3: d # d1 = p ; ::_thesis: a1,b1 // c1,d1 reconsider u1 = a, u2 = b, v1 = c, v2 = d, u = p, t1 = a1, t2 = b1, w1 = c1, w2 = d1 as VECTOR of V ; A4: ( u = u2 # t2 & u = v1 # w1 ) by A2, Def8; A5: u = v2 # w2 by A3, Def8; ( u1,u2,v1,v2 are_DTr_wrt w,y & u = u1 # t1 ) by A1, Def8, Th44; then t1,t2,w1,w2 are_DTr_wrt w,y by A4, A5, Th28; hence a1,b1 // c1,d1 by Th44; ::_thesis: verum end; Lm35: for V being RealLinearSpace for w, y being VECTOR of V for p, q, a, a1, b, b1, c, c1, d, d1 being Element of (DTrSpace (V,w,y)) st Gen w,y & p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d holds a1,b1 // c1,d1 proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V for p, q, a, a1, b, b1, c, c1, d, d1 being Element of (DTrSpace (V,w,y)) st Gen w,y & p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d holds a1,b1 // c1,d1 let w, y be VECTOR of V; ::_thesis: for p, q, a, a1, b, b1, c, c1, d, d1 being Element of (DTrSpace (V,w,y)) st Gen w,y & p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d holds a1,b1 // c1,d1 let p, q, a, a1, b, b1, c, c1, d, d1 be Element of (DTrSpace (V,w,y)); ::_thesis: ( Gen w,y & p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) assume that A1: ( Gen w,y & p <> q ) and A2: ( p,q // a,a1 & p,q // b,b1 ) and A3: ( p,q // c,c1 & p,q // d,d1 ) and A4: a,b // c,d ; ::_thesis: a1,b1 // c1,d1 reconsider u = p, u9 = q, u1 = a, u2 = b, v1 = c, v2 = d, t1 = a1, t2 = b1, w1 = c1, w2 = d1 as VECTOR of V ; A5: ( u,u9,v1,w1 are_DTr_wrt w,y & u,u9,v2,w2 are_DTr_wrt w,y ) by A3, Th44; A6: u1,u2,v1,v2 are_DTr_wrt w,y by A4, Th44; ( u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y ) by A2, Th44; then t1,t2,w1,w2 are_DTr_wrt w,y by A1, A5, A6, Th41; hence a1,b1 // c1,d1 by Th44; ::_thesis: verum end; definition let IT be non empty AfMidStruct ; attrIT is MidOrdTrapSpace-like means :Def12: :: GEOMTRAP:def 12 for a, b, c, d, a1, b1, c1, d1, p, q being Element of IT holds ( a # b = b # a & a # a = a & (a # b) # (c # d) = (a # c) # (b # d) & ex p being Element of IT st p # a = b & ( a # b = a # c implies b = c ) & ( a,b // c,d implies a,b // a # c,b # d ) & ( not a,b // c,d or a,b // a # d,b # c or a,b // b # c,a # d ) & ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) & ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of IT st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ); end; :: deftheorem Def12 defines MidOrdTrapSpace-like GEOMTRAP:def_12_:_ for IT being non empty AfMidStruct holds ( IT is MidOrdTrapSpace-like iff for a, b, c, d, a1, b1, c1, d1, p, q being Element of IT holds ( a # b = b # a & a # a = a & (a # b) # (c # d) = (a # c) # (b # d) & ex p being Element of IT st p # a = b & ( a # b = a # c implies b = c ) & ( a,b // c,d implies a,b // a # c,b # d ) & ( not a,b // c,d or a,b // a # d,b # c or a,b // b # c,a # d ) & ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) & ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of IT st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) ); registration cluster non empty strict MidOrdTrapSpace-like for AfMidStruct ; existence ex b1 being non empty AfMidStruct st ( b1 is strict & b1 is MidOrdTrapSpace-like ) proof consider V being RealLinearSpace such that A1: ex w, y being VECTOR of V st Gen w,y by ANALMETR:3; consider w, y being VECTOR of V such that A2: Gen w,y by A1; set X = DTrSpace (V,w,y); DTrSpace (V,w,y) is MidOrdTrapSpace-like proof let a, b, c, d, a1, b1, c1, d1, p, q be Element of (DTrSpace (V,w,y)); :: according to GEOMTRAP:def_12 ::_thesis: ( a # b = b # a & a # a = a & (a # b) # (c # d) = (a # c) # (b # d) & ex p being Element of (DTrSpace (V,w,y)) st p # a = b & ( a # b = a # c implies b = c ) & ( a,b // c,d implies a,b // a # c,b # d ) & ( not a,b // c,d or a,b // a # d,b # c or a,b // b # c,a # d ) & ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) & ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( a # b = b # a & a # a = a & (a # b) # (c # d) = (a # c) # (b # d) ) by Lm27, Lm28, Lm29; ::_thesis: ( ex p being Element of (DTrSpace (V,w,y)) st p # a = b & ( a # b = a # c implies b = c ) & ( a,b // c,d implies a,b // a # c,b # d ) & ( not a,b // c,d or a,b // a # d,b # c or a,b // b # c,a # d ) & ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) & ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ex p being Element of (DTrSpace (V,w,y)) st p # a = b by Lm30; ::_thesis: ( ( a # b = a # c implies b = c ) & ( a,b // c,d implies a,b // a # c,b # d ) & ( not a,b // c,d or a,b // a # d,b # c or a,b // b # c,a # d ) & ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) & ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( a # b = a # c implies b = c ) by Lm31; ::_thesis: ( ( a,b // c,d implies a,b // a # c,b # d ) & ( not a,b // c,d or a,b // a # d,b # c or a,b // b # c,a # d ) & ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) & ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( a,b // c,d implies a,b // a # c,b # d ) by A2, Lm32; ::_thesis: ( ( not a,b // c,d or a,b // a # d,b # c or a,b // b # c,a # d ) & ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) & ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( not a,b // c,d or a,b // a # d,b # c or a,b // b # c,a # d ) by A2, Lm33; ::_thesis: ( ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) & ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) by Lm34; ::_thesis: ( ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) by A2, Lm35; ::_thesis: ( ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( a,b // b,c implies ( a = b & b = c ) ) by A2, Lm22; ::_thesis: ( ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) by A2, Lm23; ::_thesis: ( ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) by Lm24; ::_thesis: ( ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) by A2, Lm25; ::_thesis: ( a,b // c,p & a,b // c,q & not a = b implies p = q ) thus ( a,b // c,p & a,b // c,q & not a = b implies p = q ) by A2, Lm26; ::_thesis: verum end; hence ex b1 being non empty AfMidStruct st ( b1 is strict & b1 is MidOrdTrapSpace-like ) ; ::_thesis: verum end; end; definition mode MidOrdTrapSpace is non empty MidOrdTrapSpace-like AfMidStruct ; end; theorem :: GEOMTRAP:46 for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds DTrSpace (V,w,y) is MidOrdTrapSpace proof let V be RealLinearSpace; ::_thesis: for w, y being VECTOR of V st Gen w,y holds DTrSpace (V,w,y) is MidOrdTrapSpace let w, y be VECTOR of V; ::_thesis: ( Gen w,y implies DTrSpace (V,w,y) is MidOrdTrapSpace ) set X = DTrSpace (V,w,y); assume A1: Gen w,y ; ::_thesis: DTrSpace (V,w,y) is MidOrdTrapSpace DTrSpace (V,w,y) is MidOrdTrapSpace-like proof let a, b, c, d, a1, b1, c1, d1, p, q be Element of (DTrSpace (V,w,y)); :: according to GEOMTRAP:def_12 ::_thesis: ( a # b = b # a & a # a = a & (a # b) # (c # d) = (a # c) # (b # d) & ex p being Element of (DTrSpace (V,w,y)) st p # a = b & ( a # b = a # c implies b = c ) & ( a,b // c,d implies a,b // a # c,b # d ) & ( not a,b // c,d or a,b // a # d,b # c or a,b // b # c,a # d ) & ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) & ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( a # b = b # a & a # a = a & (a # b) # (c # d) = (a # c) # (b # d) ) by Lm27, Lm28, Lm29; ::_thesis: ( ex p being Element of (DTrSpace (V,w,y)) st p # a = b & ( a # b = a # c implies b = c ) & ( a,b // c,d implies a,b // a # c,b # d ) & ( not a,b // c,d or a,b // a # d,b # c or a,b // b # c,a # d ) & ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) & ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ex p being Element of (DTrSpace (V,w,y)) st p # a = b by Lm30; ::_thesis: ( ( a # b = a # c implies b = c ) & ( a,b // c,d implies a,b // a # c,b # d ) & ( not a,b // c,d or a,b // a # d,b # c or a,b // b # c,a # d ) & ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) & ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( a # b = a # c implies b = c ) by Lm31; ::_thesis: ( ( a,b // c,d implies a,b // a # c,b # d ) & ( not a,b // c,d or a,b // a # d,b # c or a,b // b # c,a # d ) & ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) & ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( a,b // c,d implies a,b // a # c,b # d ) by A1, Lm32; ::_thesis: ( ( not a,b // c,d or a,b // a # d,b # c or a,b // b # c,a # d ) & ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) & ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( not a,b // c,d or a,b // a # d,b # c or a,b // b # c,a # d ) by A1, Lm33; ::_thesis: ( ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) & ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p implies a1,b1 // c1,d1 ) by Lm34; ::_thesis: ( ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) & ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) by A1, Lm35; ::_thesis: ( ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( a,b // b,c implies ( a = b & b = c ) ) by A1, Lm22; ::_thesis: ( ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) by A1, Lm23; ::_thesis: ( ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) by Lm24; ::_thesis: ( ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) thus ex d being Element of (DTrSpace (V,w,y)) st ( a,b // c,d or a,b // d,c ) by A1, Lm25; ::_thesis: ( a,b // c,p & a,b // c,q & not a = b implies p = q ) thus ( a,b // c,p & a,b // c,q & not a = b implies p = q ) by A1, Lm26; ::_thesis: verum end; hence DTrSpace (V,w,y) is MidOrdTrapSpace ; ::_thesis: verum end; set MOS = the MidOrdTrapSpace; set X = Af the MidOrdTrapSpace; Lm36: now__::_thesis:_for_a,_b,_c,_d,_a1,_b1,_c1,_d1,_p,_q_being_Element_of_(Af_the_MidOrdTrapSpace)_holds_ (_(_a,b_//_b,c_implies_(_a_=_b_&_b_=_c_)_)_&_(_a,b_//_a1,b1_&_a,b_//_c1,d1_&_a_<>_b_implies_a1,b1_//_c1,d1_)_&_(_a,b_//_c,d_implies_(_c,d_//_a,b_&_b,a_//_d,c_)_)_&_ex_d_being_Element_of_(Af_the_MidOrdTrapSpace)_st_ (_a,b_//_c,d_or_a,b_//_d,c_)_&_(_a,b_//_c,p_&_a,b_//_c,q_&_not_a_=_b_implies_p_=_q_)_) let a, b, c, d, a1, b1, c1, d1, p, q be Element of (Af the MidOrdTrapSpace); ::_thesis: ( ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (Af the MidOrdTrapSpace) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) reconsider a9 = a, b9 = b, c9 = c, d9 = d, a19 = a1, b19 = b1, c19 = c1, d19 = d1, p9 = p, q9 = q as Element of the MidOrdTrapSpace ; A1: now__::_thesis:_for_a,_b,_c,_d_being_Element_of_(Af_the_MidOrdTrapSpace) for_a9,_b9,_c9,_d9_being_Element_of_the_carrier_of_the_MidOrdTrapSpace_st_a_=_a9_&_b_=_b9_&_c_=_c9_&_d_=_d9_holds_ (_a,b_//_c,d_iff_a9,b9_//_c9,d9_) let a, b, c, d be Element of (Af the MidOrdTrapSpace); ::_thesis: for a9, b9, c9, d9 being Element of the carrier of the MidOrdTrapSpace st a = a9 & b = b9 & c = c9 & d = d9 holds ( a,b // c,d iff a9,b9 // c9,d9 ) let a9, b9, c9, d9 be Element of the carrier of the MidOrdTrapSpace; ::_thesis: ( a = a9 & b = b9 & c = c9 & d = d9 implies ( a,b // c,d iff a9,b9 // c9,d9 ) ) assume A2: ( a = a9 & b = b9 & c = c9 & d = d9 ) ; ::_thesis: ( a,b // c,d iff a9,b9 // c9,d9 ) A3: now__::_thesis:_(_a9,b9_//_c9,d9_implies_a,b_//_c,d_) assume a9,b9 // c9,d9 ; ::_thesis: a,b // c,d then [[a,b],[c,d]] in the CONGR of the MidOrdTrapSpace by A2, ANALOAF:def_2; hence a,b // c,d by ANALOAF:def_2; ::_thesis: verum end; now__::_thesis:_(_a,b_//_c,d_implies_a9,b9_//_c9,d9_) assume a,b // c,d ; ::_thesis: a9,b9 // c9,d9 then [[a,b],[c,d]] in the CONGR of the MidOrdTrapSpace by ANALOAF:def_2; hence a9,b9 // c9,d9 by A2, ANALOAF:def_2; ::_thesis: verum end; hence ( a,b // c,d iff a9,b9 // c9,d9 ) by A3; ::_thesis: verum end; hereby ::_thesis: ( ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (Af the MidOrdTrapSpace) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) assume a,b // b,c ; ::_thesis: ( a = b & b = c ) then a9,b9 // b9,c9 by A1; hence ( a = b & b = c ) by Def12; ::_thesis: verum end; hereby ::_thesis: ( ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (Af the MidOrdTrapSpace) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) assume that A4: ( a,b // a1,b1 & a,b // c1,d1 ) and A5: a <> b ; ::_thesis: a1,b1 // c1,d1 ( a9,b9 // a19,b19 & a9,b9 // c19,d19 ) by A1, A4; then a19,b19 // c19,d19 by A5, Def12; hence a1,b1 // c1,d1 by A1; ::_thesis: verum end; hereby ::_thesis: ( ex d being Element of (Af the MidOrdTrapSpace) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) assume a,b // c,d ; ::_thesis: ( c,d // a,b & b,a // d,c ) then a9,b9 // c9,d9 by A1; then ( c9,d9 // a9,b9 & b9,a9 // d9,c9 ) by Def12; hence ( c,d // a,b & b,a // d,c ) by A1; ::_thesis: verum end; thus ex d being Element of (Af the MidOrdTrapSpace) st ( a,b // c,d or a,b // d,c ) ::_thesis: ( a,b // c,p & a,b // c,q & not a = b implies p = q ) proof consider x9 being Element of the MidOrdTrapSpace such that A6: ( a9,b9 // c9,x9 or a9,b9 // x9,c9 ) by Def12; reconsider x = x9 as Element of (Af the MidOrdTrapSpace) ; take x ; ::_thesis: ( a,b // c,x or a,b // x,c ) thus ( a,b // c,x or a,b // x,c ) by A1, A6; ::_thesis: verum end; assume ( a,b // c,p & a,b // c,q ) ; ::_thesis: ( a = b or p = q ) then ( a9,b9 // c9,p9 & a9,b9 // c9,q9 ) by A1; hence ( a = b or p = q ) by Def12; ::_thesis: verum end; definition let IT be non empty AffinStruct ; attrIT is OrdTrapSpace-like means :Def13: :: GEOMTRAP:def 13 for a, b, c, d, a1, b1, c1, d1, p, q being Element of IT holds ( ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of IT st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ); end; :: deftheorem Def13 defines OrdTrapSpace-like GEOMTRAP:def_13_:_ for IT being non empty AffinStruct holds ( IT is OrdTrapSpace-like iff for a, b, c, d, a1, b1, c1, d1, p, q being Element of IT holds ( ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of IT st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) ); registration cluster non empty strict OrdTrapSpace-like for AffinStruct ; existence ex b1 being non empty AffinStruct st ( b1 is strict & b1 is OrdTrapSpace-like ) proof Af the MidOrdTrapSpace is OrdTrapSpace-like by Def13, Lm36; hence ex b1 being non empty AffinStruct st ( b1 is strict & b1 is OrdTrapSpace-like ) ; ::_thesis: verum end; end; definition mode OrdTrapSpace is non empty OrdTrapSpace-like AffinStruct ; end; registration let MOS be MidOrdTrapSpace; cluster Af MOS -> strict OrdTrapSpace-like ; coherence Af MOS is OrdTrapSpace-like proof set X = Af MOS; Af MOS is OrdTrapSpace-like proof let a, b, c, d, a1, b1, c1, d1, p, q be Element of (Af MOS); :: according to GEOMTRAP:def_13 ::_thesis: ( ( a,b // b,c implies ( a = b & b = c ) ) & ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (Af MOS) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) reconsider a9 = a, b9 = b, c9 = c, d9 = d, a19 = a1, b19 = b1, c19 = c1, d19 = d1, p9 = p, q9 = q as Element of MOS ; A1: now__::_thesis:_for_a,_b,_c,_d_being_Element_of_(Af_MOS) for_a9,_b9,_c9,_d9_being_Element_of_the_carrier_of_MOS_st_a_=_a9_&_b_=_b9_&_c_=_c9_&_d_=_d9_holds_ (_(_a,b_//_c,d_implies_a9,b9_//_c9,d9_)_&_(_a9,b9_//_c9,d9_implies_a,b_//_c,d_)_) let a, b, c, d be Element of (Af MOS); ::_thesis: for a9, b9, c9, d9 being Element of the carrier of MOS st a = a9 & b = b9 & c = c9 & d = d9 holds ( ( a,b // c,d implies a9,b9 // c9,d9 ) & ( a9,b9 // c9,d9 implies a,b // c,d ) ) let a9, b9, c9, d9 be Element of the carrier of MOS; ::_thesis: ( a = a9 & b = b9 & c = c9 & d = d9 implies ( ( a,b // c,d implies a9,b9 // c9,d9 ) & ( a9,b9 // c9,d9 implies a,b // c,d ) ) ) assume A2: ( a = a9 & b = b9 & c = c9 & d = d9 ) ; ::_thesis: ( ( a,b // c,d implies a9,b9 // c9,d9 ) & ( a9,b9 // c9,d9 implies a,b // c,d ) ) hereby ::_thesis: ( a9,b9 // c9,d9 implies a,b // c,d ) assume a,b // c,d ; ::_thesis: a9,b9 // c9,d9 then [[a,b],[c,d]] in the CONGR of MOS by ANALOAF:def_2; hence a9,b9 // c9,d9 by A2, ANALOAF:def_2; ::_thesis: verum end; assume a9,b9 // c9,d9 ; ::_thesis: a,b // c,d then [[a,b],[c,d]] in the CONGR of MOS by A2, ANALOAF:def_2; hence a,b // c,d by ANALOAF:def_2; ::_thesis: verum end; hereby ::_thesis: ( ( a,b // a1,b1 & a,b // c1,d1 & a <> b implies a1,b1 // c1,d1 ) & ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (Af MOS) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) assume a,b // b,c ; ::_thesis: ( a = b & b = c ) then a9,b9 // b9,c9 by A1; hence ( a = b & b = c ) by Def12; ::_thesis: verum end; hereby ::_thesis: ( ( a,b // c,d implies ( c,d // a,b & b,a // d,c ) ) & ex d being Element of (Af MOS) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) assume that A3: ( a,b // a1,b1 & a,b // c1,d1 ) and A4: a <> b ; ::_thesis: a1,b1 // c1,d1 ( a9,b9 // a19,b19 & a9,b9 // c19,d19 ) by A1, A3; then a19,b19 // c19,d19 by A4, Def12; hence a1,b1 // c1,d1 by A1; ::_thesis: verum end; hereby ::_thesis: ( ex d being Element of (Af MOS) st ( a,b // c,d or a,b // d,c ) & ( a,b // c,p & a,b // c,q & not a = b implies p = q ) ) assume a,b // c,d ; ::_thesis: ( c,d // a,b & b,a // d,c ) then a9,b9 // c9,d9 by A1; then ( c9,d9 // a9,b9 & b9,a9 // d9,c9 ) by Def12; hence ( c,d // a,b & b,a // d,c ) by A1; ::_thesis: verum end; thus ex d being Element of (Af MOS) st ( a,b // c,d or a,b // d,c ) ::_thesis: ( a,b // c,p & a,b // c,q & not a = b implies p = q ) proof consider x9 being Element of MOS such that A5: ( a9,b9 // c9,x9 or a9,b9 // x9,c9 ) by Def12; reconsider x = x9 as Element of (Af MOS) ; take x ; ::_thesis: ( a,b // c,x or a,b // x,c ) thus ( a,b // c,x or a,b // x,c ) by A1, A5; ::_thesis: verum end; assume ( a,b // c,p & a,b // c,q ) ; ::_thesis: ( a = b or p = q ) then ( a9,b9 // c9,p9 & a9,b9 // c9,q9 ) by A1; hence ( a = b or p = q ) by Def12; ::_thesis: verum end; hence Af MOS is OrdTrapSpace-like ; ::_thesis: verum end; end; theorem Th47: :: GEOMTRAP:47 for OTS being OrdTrapSpace for x being set holds ( x is Element of OTS iff x is Element of (Lambda OTS) ) proof let OTS be OrdTrapSpace; ::_thesis: for x being set holds ( x is Element of OTS iff x is Element of (Lambda OTS) ) let x be set ; ::_thesis: ( x is Element of OTS iff x is Element of (Lambda OTS) ) Lambda OTS = AffinStruct(# the carrier of OTS,(lambda the CONGR of OTS) #) by DIRAF:def_2; hence ( x is Element of OTS iff x is Element of (Lambda OTS) ) ; ::_thesis: verum end; theorem Th48: :: GEOMTRAP:48 for OTS being OrdTrapSpace for a, b, c, d being Element of OTS for a9, b9, c9, d9 being Element of (Lambda OTS) st a = a9 & b = b9 & c = c9 & d = d9 holds ( a9,b9 // c9,d9 iff ( a,b // c,d or a,b // d,c ) ) proof let OTS be OrdTrapSpace; ::_thesis: for a, b, c, d being Element of OTS for a9, b9, c9, d9 being Element of (Lambda OTS) st a = a9 & b = b9 & c = c9 & d = d9 holds ( a9,b9 // c9,d9 iff ( a,b // c,d or a,b // d,c ) ) let a, b, c, d be Element of OTS; ::_thesis: for a9, b9, c9, d9 being Element of (Lambda OTS) st a = a9 & b = b9 & c = c9 & d = d9 holds ( a9,b9 // c9,d9 iff ( a,b // c,d or a,b // d,c ) ) let a9, b9, c9, d9 be Element of (Lambda OTS); ::_thesis: ( a = a9 & b = b9 & c = c9 & d = d9 implies ( a9,b9 // c9,d9 iff ( a,b // c,d or a,b // d,c ) ) ) A1: Lambda OTS = AffinStruct(# the carrier of OTS,(lambda the CONGR of OTS) #) by DIRAF:def_2; assume A2: ( a = a9 & b = b9 & c = c9 & d = d9 ) ; ::_thesis: ( a9,b9 // c9,d9 iff ( a,b // c,d or a,b // d,c ) ) hereby ::_thesis: ( ( a,b // c,d or a,b // d,c ) implies a9,b9 // c9,d9 ) assume a9,b9 // c9,d9 ; ::_thesis: ( a,b // c,d or a,b // d,c ) then [[a9,b9],[c9,d9]] in lambda the CONGR of OTS by A1, ANALOAF:def_2; then ( [[a,b],[c,d]] in the CONGR of OTS or [[a,b],[d,c]] in the CONGR of OTS ) by A2, DIRAF:def_1; hence ( a,b // c,d or a,b // d,c ) by ANALOAF:def_2; ::_thesis: verum end; assume ( a,b // c,d or a,b // d,c ) ; ::_thesis: a9,b9 // c9,d9 then ( [[a,b],[c,d]] in the CONGR of OTS or [[a,b],[d,c]] in the CONGR of OTS ) by ANALOAF:def_2; then [[a,b],[c,d]] in the CONGR of (Lambda OTS) by A1, DIRAF:def_1; hence a9,b9 // c9,d9 by A2, ANALOAF:def_2; ::_thesis: verum end; Lm37: for OTS being OrdTrapSpace for a9, b9, c9 being Element of (Lambda OTS) ex d9 being Element of (Lambda OTS) st a9,b9 // c9,d9 proof let OTS be OrdTrapSpace; ::_thesis: for a9, b9, c9 being Element of (Lambda OTS) ex d9 being Element of (Lambda OTS) st a9,b9 // c9,d9 let a9, b9, c9 be Element of (Lambda OTS); ::_thesis: ex d9 being Element of (Lambda OTS) st a9,b9 // c9,d9 reconsider a = a9, b = b9, c = c9 as Element of OTS by Th47; consider d being Element of OTS such that A1: ( a,b // c,d or a,b // d,c ) by Def13; reconsider d9 = d as Element of (Lambda OTS) by Th47; take d9 ; ::_thesis: a9,b9 // c9,d9 thus a9,b9 // c9,d9 by A1, Th48; ::_thesis: verum end; Lm38: for OTS being OrdTrapSpace for a9, b9, c9, d9 being Element of (Lambda OTS) st a9,b9 // c9,d9 holds c9,d9 // a9,b9 proof let OTS be OrdTrapSpace; ::_thesis: for a9, b9, c9, d9 being Element of (Lambda OTS) st a9,b9 // c9,d9 holds c9,d9 // a9,b9 let a9, b9, c9, d9 be Element of (Lambda OTS); ::_thesis: ( a9,b9 // c9,d9 implies c9,d9 // a9,b9 ) reconsider a = a9, b = b9, c = c9, d = d9 as Element of the carrier of OTS by Th47; assume A1: a9,b9 // c9,d9 ; ::_thesis: c9,d9 // a9,b9 percases ( a,b // c,d or a,b // d,c ) by A1, Th48; suppose a,b // c,d ; ::_thesis: c9,d9 // a9,b9 then c,d // a,b by Def13; hence c9,d9 // a9,b9 by Th48; ::_thesis: verum end; suppose a,b // d,c ; ::_thesis: c9,d9 // a9,b9 then b,a // c,d by Def13; then c,d // b,a by Def13; hence c9,d9 // a9,b9 by Th48; ::_thesis: verum end; end; end; Lm39: for OTS being OrdTrapSpace for a19, b19, a9, b9, c9, d9 being Element of (Lambda OTS) st a19 <> b19 & a19,b19 // a9,b9 & a19,b19 // c9,d9 holds a9,b9 // c9,d9 proof let OTS be OrdTrapSpace; ::_thesis: for a19, b19, a9, b9, c9, d9 being Element of (Lambda OTS) st a19 <> b19 & a19,b19 // a9,b9 & a19,b19 // c9,d9 holds a9,b9 // c9,d9 let a19, b19, a9, b9, c9, d9 be Element of (Lambda OTS); ::_thesis: ( a19 <> b19 & a19,b19 // a9,b9 & a19,b19 // c9,d9 implies a9,b9 // c9,d9 ) reconsider a1 = a19, b1 = b19, a = a9, b = b9, c = c9, d = d9 as Element of OTS by Th47; assume that A1: a19 <> b19 and A2: a19,b19 // a9,b9 and A3: a19,b19 // c9,d9 ; ::_thesis: a9,b9 // c9,d9 A4: ( a1,b1 // c,d or a1,b1 // d,c ) by A3, Th48; ( a1,b1 // a,b or a1,b1 // b,a ) by A2, Th48; then ( a,b // c,d or a,b // d,c or b,a // c,d or b,a // d,c ) by A1, A4, Def13; then ( a,b // c,d or a,b // d,c ) by Def13; hence a9,b9 // c9,d9 by Th48; ::_thesis: verum end; Lm40: for OTS being OrdTrapSpace for a9, b9, c9, d9, d19 being Element of (Lambda OTS) st a9,b9 // c9,d9 & a9,b9 // c9,d19 & not a9 = b9 holds d9 = d19 proof let OTS be OrdTrapSpace; ::_thesis: for a9, b9, c9, d9, d19 being Element of (Lambda OTS) st a9,b9 // c9,d9 & a9,b9 // c9,d19 & not a9 = b9 holds d9 = d19 let a9, b9, c9, d9, d19 be Element of (Lambda OTS); ::_thesis: ( a9,b9 // c9,d9 & a9,b9 // c9,d19 & not a9 = b9 implies d9 = d19 ) reconsider a = a9, b = b9, c = c9, d = d9, d1 = d19 as Element of OTS by Th47; assume ( a9,b9 // c9,d9 & a9,b9 // c9,d19 ) ; ::_thesis: ( a9 = b9 or d9 = d19 ) then A1: ( ( a,b // c,d & a,b // c,d1 ) or ( a,b // c,d & a,b // d1,c ) or ( a,b // d,c & a,b // c,d1 ) or ( a,b // d,c & a,b // d1,c ) ) by Th48; assume A2: a9 <> b9 ; ::_thesis: d9 = d19 then ( d = d1 or d1,c // c,d or d,c // c,d1 or ( b,a // c,d & b,a // c,d1 ) ) by A1, Def13; then ( d = d1 or ( c = d & c = d1 ) ) by A2, Def13; hence d9 = d19 ; ::_thesis: verum end; Lm41: for OTS being OrdTrapSpace for a, b being Element of OTS holds a,b // a,b proof let OTS be OrdTrapSpace; ::_thesis: for a, b being Element of OTS holds a,b // a,b let a, b be Element of OTS; ::_thesis: a,b // a,b consider c being Element of OTS such that A1: ( a,b // a,c or a,b // c,a ) by Def13; percases ( a,b // c,a or a,b // a,c ) by A1; suppose a,b // c,a ; ::_thesis: a,b // a,b then A2: c,a // a,b by Def13; then c = a by Def13; hence a,b // a,b by A2, Def13; ::_thesis: verum end; supposeA3: a,b // a,c ; ::_thesis: a,b // a,b percases ( a <> c or a = c ) ; supposeA4: a <> c ; ::_thesis: a,b // a,b a,c // a,b by A3, Def13; hence a,b // a,b by A4, Def13; ::_thesis: verum end; suppose a = c ; ::_thesis: a,b // a,b then a,a // a,b by A3, Def13; hence a,b // a,b by Def13; ::_thesis: verum end; end; end; end; end; Lm42: for OTS being OrdTrapSpace for a9, b9 being Element of (Lambda OTS) holds a9,b9 // b9,a9 proof let OTS be OrdTrapSpace; ::_thesis: for a9, b9 being Element of (Lambda OTS) holds a9,b9 // b9,a9 let a9, b9 be Element of (Lambda OTS); ::_thesis: a9,b9 // b9,a9 reconsider a = a9, b = b9 as Element of OTS by Th47; a,b // a,b by Lm41; hence a9,b9 // b9,a9 by Th48; ::_thesis: verum end; definition let IT be non empty AffinStruct ; attrIT is TrapSpace-like means :: GEOMTRAP:def 14 for a9, b9, c9, d9, p9, q9 being Element of IT holds ( a9,b9 // b9,a9 & ( a9,b9 // c9,d9 & a9,b9 // c9,q9 & not a9 = b9 implies d9 = q9 ) & ( p9 <> q9 & p9,q9 // a9,b9 & p9,q9 // c9,d9 implies a9,b9 // c9,d9 ) & ( a9,b9 // c9,d9 implies c9,d9 // a9,b9 ) & ex x9 being Element of IT st a9,b9 // c9,x9 ); end; :: deftheorem defines TrapSpace-like GEOMTRAP:def_14_:_ for IT being non empty AffinStruct holds ( IT is TrapSpace-like iff for a9, b9, c9, d9, p9, q9 being Element of IT holds ( a9,b9 // b9,a9 & ( a9,b9 // c9,d9 & a9,b9 // c9,q9 & not a9 = b9 implies d9 = q9 ) & ( p9 <> q9 & p9,q9 // a9,b9 & p9,q9 // c9,d9 implies a9,b9 // c9,d9 ) & ( a9,b9 // c9,d9 implies c9,d9 // a9,b9 ) & ex x9 being Element of IT st a9,b9 // c9,x9 ) ); Lm43: for OTS being OrdTrapSpace holds Lambda OTS is TrapSpace-like proof let OTS be OrdTrapSpace; ::_thesis: Lambda OTS is TrapSpace-like set TS = Lambda OTS; let a9, b9, c9, d9, p9, q9 be Element of (Lambda OTS); :: according to GEOMTRAP:def_14 ::_thesis: ( a9,b9 // b9,a9 & ( a9,b9 // c9,d9 & a9,b9 // c9,q9 & not a9 = b9 implies d9 = q9 ) & ( p9 <> q9 & p9,q9 // a9,b9 & p9,q9 // c9,d9 implies a9,b9 // c9,d9 ) & ( a9,b9 // c9,d9 implies c9,d9 // a9,b9 ) & ex x9 being Element of (Lambda OTS) st a9,b9 // c9,x9 ) thus a9,b9 // b9,a9 by Lm42; ::_thesis: ( ( a9,b9 // c9,d9 & a9,b9 // c9,q9 & not a9 = b9 implies d9 = q9 ) & ( p9 <> q9 & p9,q9 // a9,b9 & p9,q9 // c9,d9 implies a9,b9 // c9,d9 ) & ( a9,b9 // c9,d9 implies c9,d9 // a9,b9 ) & ex x9 being Element of (Lambda OTS) st a9,b9 // c9,x9 ) thus ( a9,b9 // c9,d9 & a9,b9 // c9,q9 & not a9 = b9 implies d9 = q9 ) by Lm40; ::_thesis: ( ( p9 <> q9 & p9,q9 // a9,b9 & p9,q9 // c9,d9 implies a9,b9 // c9,d9 ) & ( a9,b9 // c9,d9 implies c9,d9 // a9,b9 ) & ex x9 being Element of (Lambda OTS) st a9,b9 // c9,x9 ) thus ( p9 <> q9 & p9,q9 // a9,b9 & p9,q9 // c9,d9 implies a9,b9 // c9,d9 ) by Lm39; ::_thesis: ( ( a9,b9 // c9,d9 implies c9,d9 // a9,b9 ) & ex x9 being Element of (Lambda OTS) st a9,b9 // c9,x9 ) thus ( a9,b9 // c9,d9 implies c9,d9 // a9,b9 ) by Lm38; ::_thesis: ex x9 being Element of (Lambda OTS) st a9,b9 // c9,x9 thus ex x9 being Element of (Lambda OTS) st a9,b9 // c9,x9 by Lm37; ::_thesis: verum end; registration cluster non empty strict TrapSpace-like for AffinStruct ; existence ex b1 being non empty AffinStruct st ( b1 is strict & b1 is TrapSpace-like ) proof set TS = Lambda the OrdTrapSpace; Lambda the OrdTrapSpace is TrapSpace-like by Lm43; hence ex b1 being non empty AffinStruct st ( b1 is strict & b1 is TrapSpace-like ) ; ::_thesis: verum end; end; definition mode TrapSpace is non empty TrapSpace-like AffinStruct ; end; registration let OTS be OrdTrapSpace; cluster Lambda OTS -> TrapSpace-like ; correctness coherence Lambda OTS is TrapSpace-like ; by Lm43; end; definition let IT be non empty AffinStruct ; attrIT is Regular means :Def15: :: GEOMTRAP:def 15 for p, q, a, a1, b, b1, c, c1, d, d1 being Element of IT st p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d holds a1,b1 // c1,d1; end; :: deftheorem Def15 defines Regular GEOMTRAP:def_15_:_ for IT being non empty AffinStruct holds ( IT is Regular iff for p, q, a, a1, b, b1, c, c1, d, d1 being Element of IT st p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d holds a1,b1 // c1,d1 ); registration cluster non empty strict OrdTrapSpace-like Regular for AffinStruct ; existence ex b1 being OrdTrapSpace st ( b1 is strict & b1 is Regular ) proof set MOTS = the MidOrdTrapSpace; set X = Af the MidOrdTrapSpace; A1: now__::_thesis:_for_a,_b,_c,_d_being_Element_of_(Af_the_MidOrdTrapSpace) for_a9,_b9,_c9,_d9_being_Element_of_the_carrier_of_the_MidOrdTrapSpace_st_a_=_a9_&_b_=_b9_&_c_=_c9_&_d_=_d9_holds_ (_(_a,b_//_c,d_implies_a9,b9_//_c9,d9_)_&_(_a9,b9_//_c9,d9_implies_a,b_//_c,d_)_) let a, b, c, d be Element of (Af the MidOrdTrapSpace); ::_thesis: for a9, b9, c9, d9 being Element of the carrier of the MidOrdTrapSpace st a = a9 & b = b9 & c = c9 & d = d9 holds ( ( a,b // c,d implies a9,b9 // c9,d9 ) & ( a9,b9 // c9,d9 implies a,b // c,d ) ) let a9, b9, c9, d9 be Element of the carrier of the MidOrdTrapSpace; ::_thesis: ( a = a9 & b = b9 & c = c9 & d = d9 implies ( ( a,b // c,d implies a9,b9 // c9,d9 ) & ( a9,b9 // c9,d9 implies a,b // c,d ) ) ) assume A2: ( a = a9 & b = b9 & c = c9 & d = d9 ) ; ::_thesis: ( ( a,b // c,d implies a9,b9 // c9,d9 ) & ( a9,b9 // c9,d9 implies a,b // c,d ) ) hereby ::_thesis: ( a9,b9 // c9,d9 implies a,b // c,d ) assume a,b // c,d ; ::_thesis: a9,b9 // c9,d9 then [[a,b],[c,d]] in the CONGR of the MidOrdTrapSpace by ANALOAF:def_2; hence a9,b9 // c9,d9 by A2, ANALOAF:def_2; ::_thesis: verum end; assume a9,b9 // c9,d9 ; ::_thesis: a,b // c,d then [[a,b],[c,d]] in the CONGR of the MidOrdTrapSpace by A2, ANALOAF:def_2; hence a,b // c,d by ANALOAF:def_2; ::_thesis: verum end; Af the MidOrdTrapSpace is Regular proof let p, q, a, a1, b, b1, c, c1, d, d1 be Element of (Af the MidOrdTrapSpace); :: according to GEOMTRAP:def_15 ::_thesis: ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) assume that A3: p <> q and A4: ( p,q // a,a1 & p,q // b,b1 ) and A5: ( p,q // c,c1 & p,q // d,d1 ) and A6: a,b // c,d ; ::_thesis: a1,b1 // c1,d1 reconsider a9 = a, b9 = b, c9 = c, d9 = d, a19 = a1, b19 = b1, c19 = c1, d19 = d1, p9 = p, q9 = q as Element of the MidOrdTrapSpace ; A7: ( p9,q9 // c9,c19 & p9,q9 // d9,d19 ) by A1, A5; A8: a9,b9 // c9,d9 by A1, A6; ( p9,q9 // a9,a19 & p9,q9 // b9,b19 ) by A1, A4; then a19,b19 // c19,d19 by A3, A7, A8, Def12; hence a1,b1 // c1,d1 by A1; ::_thesis: verum end; hence ex b1 being OrdTrapSpace st ( b1 is strict & b1 is Regular ) ; ::_thesis: verum end; end; registration let MOS be MidOrdTrapSpace; cluster Af MOS -> strict Regular ; correctness coherence Af MOS is Regular ; proof set X = Af MOS; now__::_thesis:_for_p,_q,_a,_a1,_b,_b1,_c,_c1,_d,_d1_being_Element_of_(Af_MOS)_st_p_<>_q_&_p,q_//_a,a1_&_p,q_//_b,b1_&_p,q_//_c,c1_&_p,q_//_d,d1_&_a,b_//_c,d_holds_ a1,b1_//_c1,d1 let p, q, a, a1, b, b1, c, c1, d, d1 be Element of (Af MOS); ::_thesis: ( p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d implies a1,b1 // c1,d1 ) assume that A1: p <> q and A2: ( p,q // a,a1 & p,q // b,b1 ) and A3: ( p,q // c,c1 & p,q // d,d1 ) and A4: a,b // c,d ; ::_thesis: a1,b1 // c1,d1 reconsider a9 = a, b9 = b, c9 = c, d9 = d, a19 = a1, b19 = b1, c19 = c1, d19 = d1, p9 = p, q9 = q as Element of MOS ; A5: now__::_thesis:_for_a,_b,_c,_d_being_Element_of_(Af_MOS) for_a9,_b9,_c9,_d9_being_Element_of_the_carrier_of_MOS_st_a_=_a9_&_b_=_b9_&_c_=_c9_&_d_=_d9_holds_ (_(_a,b_//_c,d_implies_a9,b9_//_c9,d9_)_&_(_a9,b9_//_c9,d9_implies_a,b_//_c,d_)_) let a, b, c, d be Element of (Af MOS); ::_thesis: for a9, b9, c9, d9 being Element of the carrier of MOS st a = a9 & b = b9 & c = c9 & d = d9 holds ( ( a,b // c,d implies a9,b9 // c9,d9 ) & ( a9,b9 // c9,d9 implies a,b // c,d ) ) let a9, b9, c9, d9 be Element of the carrier of MOS; ::_thesis: ( a = a9 & b = b9 & c = c9 & d = d9 implies ( ( a,b // c,d implies a9,b9 // c9,d9 ) & ( a9,b9 // c9,d9 implies a,b // c,d ) ) ) assume A6: ( a = a9 & b = b9 & c = c9 & d = d9 ) ; ::_thesis: ( ( a,b // c,d implies a9,b9 // c9,d9 ) & ( a9,b9 // c9,d9 implies a,b // c,d ) ) hereby ::_thesis: ( a9,b9 // c9,d9 implies a,b // c,d ) assume a,b // c,d ; ::_thesis: a9,b9 // c9,d9 then [[a,b],[c,d]] in the CONGR of MOS by ANALOAF:def_2; hence a9,b9 // c9,d9 by A6, ANALOAF:def_2; ::_thesis: verum end; assume a9,b9 // c9,d9 ; ::_thesis: a,b // c,d then [[a,b],[c,d]] in the CONGR of MOS by A6, ANALOAF:def_2; hence a,b // c,d by ANALOAF:def_2; ::_thesis: verum end; then A7: ( p9,q9 // c9,c19 & p9,q9 // d9,d19 ) by A3; A8: a9,b9 // c9,d9 by A4, A5; ( p9,q9 // a9,a19 & p9,q9 // b9,b19 ) by A2, A5; then a19,b19 // c19,d19 by A1, A7, A8, Def12; hence a1,b1 // c1,d1 by A5; ::_thesis: verum end; hence Af MOS is Regular by Def15; ::_thesis: verum end; end;