:: EUCLID_8 semantic presentation begin definition let x, y, z be real number ; :: original: <* redefine func|[x,y,z]| -> Element of REAL 3; coherence <*x,y,z*> is Element of REAL 3 proof reconsider x = x, y = y, z = z as Real by XREAL_0:def_1; <*x,y,z*> is Element of REAL 3 by FINSEQ_2:104; hence <*x,y,z*> is Element of REAL 3 ; ::_thesis: verum end; end; theorem Th1: :: EUCLID_8:1 for p being Element of REAL 3 holds p = |[(p . 1),(p . 2),(p . 3)]| proof let p be Element of REAL 3; ::_thesis: p = |[(p . 1),(p . 2),(p . 3)]| consider x, y, z being Element of REAL such that A1: p = <*x,y,z*> by FINSEQ_2:103; A2: p . 1 = x by A1, FINSEQ_1:45; p . 2 = y by A1, FINSEQ_1:45; hence p = |[(p . 1),(p . 2),(p . 3)]| by A1, A2, FINSEQ_1:45; ::_thesis: verum end; Lm1: for p being Element of REAL 3 for r being real number holds r * p = |[(r * (p . 1)),(r * (p . 2)),(r * (p . 3))]| proof let p be Element of REAL 3; ::_thesis: for r being real number holds r * p = |[(r * (p . 1)),(r * (p . 2)),(r * (p . 3))]| let r be real number ; ::_thesis: r * p = |[(r * (p . 1)),(r * (p . 2)),(r * (p . 3))]| A1: (r * p) . 1 = r * (p . 1) by RVSUM_1:44; A2: (r * p) . 2 = r * (p . 2) by RVSUM_1:44; (r * p) . 3 = r * (p . 3) by RVSUM_1:44; hence r * p = |[(r * (p . 1)),(r * (p . 2)),(r * (p . 3))]| by A1, A2, Th1; ::_thesis: verum end; Lm2: for p1, p2 being Element of REAL 3 holds p1 + p2 = |[((p1 . 1) + (p2 . 1)),((p1 . 2) + (p2 . 2)),((p1 . 3) + (p2 . 3))]| proof let p1, p2 be Element of REAL 3; ::_thesis: p1 + p2 = |[((p1 . 1) + (p2 . 1)),((p1 . 2) + (p2 . 2)),((p1 . 3) + (p2 . 3))]| A1: (p1 + p2) . 1 = (p1 . 1) + (p2 . 1) by RVSUM_1:11; A2: (p1 + p2) . 2 = (p1 . 2) + (p2 . 2) by RVSUM_1:11; (p1 + p2) . 3 = (p1 . 3) + (p2 . 3) by RVSUM_1:11; hence p1 + p2 = |[((p1 . 1) + (p2 . 1)),((p1 . 2) + (p2 . 2)),((p1 . 3) + (p2 . 3))]| by A1, A2, Th1; ::_thesis: verum end; Lm3: for p being Element of REAL 3 holds ( (- p) . 1 = - (p . 1) & (- p) . 2 = - (p . 2) & (- p) . 3 = - (p . 3) ) proof let p be Element of REAL 3; ::_thesis: ( (- p) . 1 = - (p . 1) & (- p) . 2 = - (p . 2) & (- p) . 3 = - (p . 3) ) - p = |[((- 1) * (p . 1)),((- 1) * (p . 2)),((- 1) * (p . 3))]| by Lm1 .= |[(- (p . 1)),(- (p . 2)),(- (p . 3))]| ; hence ( (- p) . 1 = - (p . 1) & (- p) . 2 = - (p . 2) & (- p) . 3 = - (p . 3) ) by FINSEQ_1:45; ::_thesis: verum end; theorem :: EUCLID_8:2 for f being FinSequence of REAL st len f = 3 holds f is Element of REAL 3 proof let f be FinSequence of REAL ; ::_thesis: ( len f = 3 implies f is Element of REAL 3 ) assume A1: len f = 3 ; ::_thesis: f is Element of REAL 3 consider x1, x2, x3 being Element of REAL such that A2: ( f . 1 = x1 & f . 2 = x2 & f . 3 = x3 ) ; <*x1,x2,x3*> is Element of 3 -tuples_on REAL by FINSEQ_2:104; hence f is Element of REAL 3 by A1, A2, FINSEQ_1:45; ::_thesis: verum end; Lm4: for p1, p2 being Element of REAL 3 holds p1 - p2 = |[((p1 . 1) - (p2 . 1)),((p1 . 2) - (p2 . 2)),((p1 . 3) - (p2 . 3))]| proof let p1, p2 be Element of REAL 3; ::_thesis: p1 - p2 = |[((p1 . 1) - (p2 . 1)),((p1 . 2) - (p2 . 2)),((p1 . 3) - (p2 . 3))]| A1: (- p2) . 1 = - (p2 . 1) by Lm3; A2: (- p2) . 2 = - (p2 . 2) by Lm3; (- p2) . 3 = - (p2 . 3) by Lm3; then p1 + (- p2) = |[((p1 . 1) + (- (p2 . 1))),((p1 . 2) + (- (p2 . 2))),((p1 . 3) + (- (p2 . 3)))]| by A1, A2, Lm2; hence p1 - p2 = |[((p1 . 1) - (p2 . 1)),((p1 . 2) - (p2 . 2)),((p1 . 3) - (p2 . 3))]| ; ::_thesis: verum end; Lm5: for p1, p2 being Element of REAL 3 holds |(p1,p2)| = (((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (p2 . 3)) proof let p1, p2 be Element of REAL 3; ::_thesis: |(p1,p2)| = (((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (p2 . 3)) reconsider f1 = p1, f2 = p2 as FinSequence of REAL ; A1: len f1 = len <*(p1 . 1),(p1 . 2),(p1 . 3)*> by Th1 .= 3 by FINSEQ_1:45 ; len f2 = len <*(p2 . 1),(p2 . 2),(p2 . 3)*> by Th1 .= 3 by FINSEQ_1:45 ; then |(p1,p2)| = Sum <*((f1 . 1) * (f2 . 1)),((f1 . 2) * (f2 . 2)),((f1 . 3) * (f2 . 3))*> by A1, EUCLID_5:28 .= (((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (f2 . 3)) by RVSUM_1:78 ; hence |(p1,p2)| = (((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (p2 . 3)) ; ::_thesis: verum end; Lm6: for r, x, y, z being Element of REAL holds r * |[x,y,z]| = |[(r * x),(r * y),(r * z)]| proof let r, x, y, z be Element of REAL ; ::_thesis: r * |[x,y,z]| = |[(r * x),(r * y),(r * z)]| set p = |[x,y,z]|; r * |[x,y,z]| = |[(r * (|[x,y,z]| . 1)),(r * (|[x,y,z]| . 2)),(r * (|[x,y,z]| . 3))]| by Lm1 .= |[(r * x),(r * (|[x,y,z]| . 2)),(r * (|[x,y,z]| . 3))]| by FINSEQ_1:45 .= |[(r * x),(r * y),(r * (|[x,y,z]| . 3))]| by FINSEQ_1:45 .= |[(r * x),(r * y),(r * z)]| by FINSEQ_1:45 ; hence r * |[x,y,z]| = |[(r * x),(r * y),(r * z)]| ; ::_thesis: verum end; Lm7: for p1, p2 being Element of REAL 3 holds p1 + (- p2) = |[((p1 . 1) - (p2 . 1)),((p1 . 2) - (p2 . 2)),((p1 . 3) - (p2 . 3))]| proof let p1, p2 be Element of REAL 3; ::_thesis: p1 + (- p2) = |[((p1 . 1) - (p2 . 1)),((p1 . 2) - (p2 . 2)),((p1 . 3) - (p2 . 3))]| A1: - p2 = |[((- 1) * (p2 . 1)),((- 1) * (p2 . 2)),((- 1) * (p2 . 3))]| by Lm1 .= |[(- (p2 . 1)),(- (p2 . 2)),(- (p2 . 3))]| ; p1 + (- p2) = |[((p1 . 1) + ((- p2) . 1)),((p1 . 2) + ((- p2) . 2)),((p1 . 3) + ((- p2) . 3))]| by Lm2 .= |[((p1 . 1) + (- (p2 . 1))),((p1 . 2) + ((- p2) . 2)),((p1 . 3) + ((- p2) . 3))]| by A1, FINSEQ_1:45 .= |[((p1 . 1) + (- (p2 . 1))),((p1 . 2) + (- (p2 . 2))),((p1 . 3) + ((- p2) . 3))]| by A1, FINSEQ_1:45 .= |[((p1 . 1) + (- (p2 . 1))),((p1 . 2) + (- (p2 . 2))),((p1 . 3) + (- (p2 . 3)))]| by A1, FINSEQ_1:45 ; hence p1 + (- p2) = |[((p1 . 1) - (p2 . 1)),((p1 . 2) - (p2 . 2)),((p1 . 3) - (p2 . 3))]| ; ::_thesis: verum end; Lm8: for x1, x2, x3, y1, y2, y3 being Element of REAL holds |[x1,x2,x3]| + |[y1,y2,y3]| = |[(x1 + y1),(x2 + y2),(x3 + y3)]| proof let x1, x2, x3, y1, y2, y3 be Element of REAL ; ::_thesis: |[x1,x2,x3]| + |[y1,y2,y3]| = |[(x1 + y1),(x2 + y2),(x3 + y3)]| A1: |[y1,y2,y3]| . 1 = y1 by FINSEQ_1:45; A2: |[y1,y2,y3]| . 2 = y2 by FINSEQ_1:45; A3: |[y1,y2,y3]| . 3 = y3 by FINSEQ_1:45; A4: (|[x1,x2,x3]| . 1) + (|[y1,y2,y3]| . 1) = x1 + y1 by A1, FINSEQ_1:45; A5: (|[x1,x2,x3]| . 2) + (|[y1,y2,y3]| . 2) = x2 + y2 by A2, FINSEQ_1:45; (|[x1,x2,x3]| . 3) + (|[y1,y2,y3]| . 3) = x3 + y3 by A3, FINSEQ_1:45; hence |[x1,x2,x3]| + |[y1,y2,y3]| = |[(x1 + y1),(x2 + y2),(x3 + y3)]| by A4, A5, Lm2; ::_thesis: verum end; Lm9: for p1, p2, q1, q2 being Element of REAL 3 holds (p1 + p2) + (q1 + q2) = (p1 + q1) + (p2 + q2) proof let p1, p2, q1, q2 be Element of REAL 3; ::_thesis: (p1 + p2) + (q1 + q2) = (p1 + q1) + (p2 + q2) thus (p1 + p2) + (q1 + q2) = ((p1 + p2) + q1) + q2 by RVSUM_1:15 .= ((p1 + q1) + p2) + q2 by RVSUM_1:15 .= (p1 + q1) + (p2 + q2) by RVSUM_1:15 ; ::_thesis: verum end; Lm10: for p1, p2, q1, q2 being Element of REAL 3 holds (p1 + p2) - (q1 + q2) = (p1 - q1) + (p2 - q2) proof let p1, p2, q1, q2 be Element of REAL 3; ::_thesis: (p1 + p2) - (q1 + q2) = (p1 - q1) + (p2 - q2) thus (p1 + p2) - (q1 + q2) = ((p1 + p2) - q1) - q2 by RVSUM_1:39 .= (p1 + p2) + ((- q1) + (- q2)) by RVSUM_1:15 .= (p1 - q1) + (p2 - q2) by Lm9 ; ::_thesis: verum end; Lm11: for x1, x2, x3, y1, y2, y3 being Element of REAL holds |[x1,x2,x3]| - |[y1,y2,y3]| = |[(x1 - y1),(x2 - y2),(x3 - y3)]| proof let x1, x2, x3, y1, y2, y3 be Element of REAL ; ::_thesis: |[x1,x2,x3]| - |[y1,y2,y3]| = |[(x1 - y1),(x2 - y2),(x3 - y3)]| A1: |[y1,y2,y3]| . 1 = y1 by FINSEQ_1:45; A2: |[y1,y2,y3]| . 2 = y2 by FINSEQ_1:45; A3: |[y1,y2,y3]| . 3 = y3 by FINSEQ_1:45; A4: (|[x1,x2,x3]| . 1) - (|[y1,y2,y3]| . 1) = x1 - y1 by A1, FINSEQ_1:45; A5: (|[x1,x2,x3]| . 2) - (|[y1,y2,y3]| . 2) = x2 - y2 by A2, FINSEQ_1:45; (|[x1,x2,x3]| . 3) - (|[y1,y2,y3]| . 3) = x3 - y3 by A3, FINSEQ_1:45; hence |[x1,x2,x3]| - |[y1,y2,y3]| = |[(x1 - y1),(x2 - y2),(x3 - y3)]| by A4, A5, Lm7; ::_thesis: verum end; definition func -> Element of REAL 3 equals :: EUCLID_8:def 1 |[1,0,0]|; coherence |[1,0,0]| is Element of REAL 3 ; func -> Element of REAL 3 equals :: EUCLID_8:def 2 |[0,1,0]|; coherence |[0,1,0]| is Element of REAL 3 ; func -> Element of REAL 3 equals :: EUCLID_8:def 3 |[0,0,1]|; coherence |[0,0,1]| is Element of REAL 3 ; end; :: deftheorem defines EUCLID_8:def_1_:_ = |[1,0,0]|; :: deftheorem defines EUCLID_8:def_2_:_ = |[0,1,0]|; :: deftheorem defines EUCLID_8:def_3_:_ = |[0,0,1]|; Lm12: for x, y, z being Element of REAL for p being Element of REAL 3 st p = |[x,y,z]| holds |(p,p)| = ((x ^2) + (y ^2)) + (z ^2) proof let x, y, z be Element of REAL ; ::_thesis: for p being Element of REAL 3 st p = |[x,y,z]| holds |(p,p)| = ((x ^2) + (y ^2)) + (z ^2) let p be Element of REAL 3; ::_thesis: ( p = |[x,y,z]| implies |(p,p)| = ((x ^2) + (y ^2)) + (z ^2) ) assume p = |[x,y,z]| ; ::_thesis: |(p,p)| = ((x ^2) + (y ^2)) + (z ^2) then ( p . 1 = x & p . 2 = y & p . 3 = z ) by FINSEQ_1:45; hence |(p,p)| = ((x ^2) + (y ^2)) + (z ^2) by Lm5; ::_thesis: verum end; definition let p1, p2 be Element of REAL 3; funcp1 p2 -> Element of REAL 3 equals :: EUCLID_8:def 4 |[(((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2))),(((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))),(((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1)))]|; correctness coherence |[(((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2))),(((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))),(((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1)))]| is Element of REAL 3; ; end; :: deftheorem defines EUCLID_8:def_4_:_ for p1, p2 being Element of REAL 3 holds p1 p2 = |[(((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2))),(((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))),(((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1)))]|; Lm13: for x1, x2, x3, y1, y2, y3 being Element of REAL holds |[x1,x2,x3]| |[y1,y2,y3]| = |[((x2 * y3) - (x3 * y2)),((x3 * y1) - (x1 * y3)),((x1 * y2) - (x2 * y1))]| proof let x1, x2, x3, y1, y2, y3 be Element of REAL ; ::_thesis: |[x1,x2,x3]| |[y1,y2,y3]| = |[((x2 * y3) - (x3 * y2)),((x3 * y1) - (x1 * y3)),((x1 * y2) - (x2 * y1))]| set p1 = |[x1,x2,x3]|; A1: ( |[x1,x2,x3]| . 1 = x1 & |[x1,x2,x3]| . 2 = x2 & |[x1,x2,x3]| . 3 = x3 ) by FINSEQ_1:45; set p2 = |[y1,y2,y3]|; ( |[y1,y2,y3]| . 1 = y1 & |[y1,y2,y3]| . 2 = y2 & |[y1,y2,y3]| . 3 = y3 ) by FINSEQ_1:45; hence |[x1,x2,x3]| |[y1,y2,y3]| = |[((x2 * y3) - (x3 * y2)),((x3 * y1) - (x1 * y3)),((x1 * y2) - (x2 * y1))]| by A1; ::_thesis: verum end; Lm14: for r being Element of REAL for p1, p2 being Element of REAL 3 holds ( (r * p1) p2 = r * (p1 p2) & (r * p1) p2 = p1 (r * p2) ) proof let r be Element of REAL ; ::_thesis: for p1, p2 being Element of REAL 3 holds ( (r * p1) p2 = r * (p1 p2) & (r * p1) p2 = p1 (r * p2) ) let p1, p2 be Element of REAL 3; ::_thesis: ( (r * p1) p2 = r * (p1 p2) & (r * p1) p2 = p1 (r * p2) ) A1: ( (p1 p2) . 1 = ((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2)) & (p1 p2) . 2 = ((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3)) & (p1 p2) . 3 = ((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1)) ) by FINSEQ_1:45; A2: (r * p1) p2 = |[(r * (p1 . 1)),(r * (p1 . 2)),(r * (p1 . 3))]| p2 by Lm1 .= |[(r * (p1 . 1)),(r * (p1 . 2)),(r * (p1 . 3))]| |[(p2 . 1),(p2 . 2),(p2 . 3)]| by Th1 .= |[(((r * (p1 . 2)) * (p2 . 3)) - ((r * (p1 . 3)) * (p2 . 2))),(((r * (p1 . 3)) * (p2 . 1)) - ((r * (p1 . 1)) * (p2 . 3))),(((r * (p1 . 1)) * (p2 . 2)) - ((r * (p1 . 2)) * (p2 . 1)))]| by Lm13 ; then A3: (r * p1) p2 = |[(r * ((p1 p2) . 1)),(r * ((p1 p2) . 2)),(r * ((p1 p2) . 3))]| by A1 .= r * (p1 p2) by Lm1 ; (r * p1) p2 = |[(((p1 . 2) * (r * (p2 . 3))) - ((p1 . 3) * (r * (p2 . 2)))),(((p1 . 3) * (r * (p2 . 1))) - ((p1 . 1) * (r * (p2 . 3)))),(((p1 . 1) * (r * (p2 . 2))) - ((p1 . 2) * (r * (p2 . 1))))]| by A2 .= |[(p1 . 1),(p1 . 2),(p1 . 3)]| |[(r * (p2 . 1)),(r * (p2 . 2)),(r * (p2 . 3))]| by Lm13 .= p1 |[(r * (p2 . 1)),(r * (p2 . 2)),(r * (p2 . 3))]| by Th1 .= p1 (r * p2) by Lm1 ; hence ( (r * p1) p2 = r * (p1 p2) & (r * p1) p2 = p1 (r * p2) ) by A3; ::_thesis: verum end; theorem :: EUCLID_8:3 for p1, p2 being Element of REAL 3 st p1,p2 are_ldependent2 holds p1 p2 = 0.REAL 3 proof let p1, p2 be Element of REAL 3; ::_thesis: ( p1,p2 are_ldependent2 implies p1 p2 = 0.REAL 3 ) assume p1,p2 are_ldependent2 ; ::_thesis: p1 p2 = 0.REAL 3 then A1: ex a1, a2 being Real st ( (a1 * p1) + (a2 * p2) = 0.REAL 3 & ( a1 <> 0 or a2 <> 0 ) ) by EUCLIDLP:def_2; now__::_thesis:_(_(_ex_a1,_a2_being_Real_st_ (_(a1_*_p1)_+_(a2_*_p2)_=_0.REAL_3_&_a1_<>_0_)_&_p1__p2_=_0.REAL_3_)_or_(_ex_a1,_a2_being_Real_st_ (_(a1_*_p1)_+_(a2_*_p2)_=_0.REAL_3_&_a2_<>_0_)_&_p1__p2_=_0.REAL_3_)_) percases ( ex a1, a2 being Real st ( (a1 * p1) + (a2 * p2) = 0.REAL 3 & a1 <> 0 ) or ex a1, a2 being Real st ( (a1 * p1) + (a2 * p2) = 0.REAL 3 & a2 <> 0 ) ) by A1; case ex a1, a2 being Real st ( (a1 * p1) + (a2 * p2) = 0.REAL 3 & a1 <> 0 ) ; ::_thesis: p1 p2 = 0.REAL 3 then consider a1, a2 being Real such that A2: ( a1 <> 0 & (a1 * p1) + (a2 * p2) = 0.REAL 3 ) ; A3: (1 / a1) * (a1 * p1) = (1 / a1) * (- (a2 * p2)) by A2, RVSUM_1:23 .= (1 / a1) * (((- 1) * a2) * p2) by RVSUM_1:49 .= (1 / a1) * ((- a2) * p2) ; A4: (1 / a1) * (a1 * p1) = (a1 * (1 / a1)) * p1 by EUCLID_4:4 .= 1 * p1 by A2, XCMPLX_1:106 ; A5: 1 * p1 = |[(1 * (p1 . 1)),(1 * (p1 . 2)),(1 * (p1 . 3))]| by Lm1 .= p1 by Th1 ; A6: 0.REAL 3 = |[0,0,0]| by FINSEQ_2:62; p1 p2 = (((- a2) * (1 / a1)) * p2) p2 by A3, A4, A5, EUCLID_4:4 .= (((- a2) / a1) * p2) p2 by XCMPLX_1:99 ; then p1 p2 = ((- a2) / a1) * (p2 p2) by Lm14; then p1 p2 = ((- a2) / a1) * (0.REAL 3) by FINSEQ_2:62 .= |[(((- a2) / a1) * ((0.REAL 3) . 1)),(((- a2) / a1) * ((0.REAL 3) . 2)),(((- a2) / a1) * ((0.REAL 3) . 3))]| by Lm1 .= |[(((- a2) / a1) * 0),(((- a2) / a1) * ((0.REAL 3) . 2)),(((- a2) / a1) * ((0.REAL 3) . 3))]| by A6, FINSEQ_1:45 .= |[0,(((- a2) / a1) * 0),(((- a2) / a1) * ((0.REAL 3) . 3))]| by A6, FINSEQ_1:45 .= 0.REAL 3 by A6, FINSEQ_1:45 ; hence p1 p2 = 0.REAL 3 ; ::_thesis: verum end; case ex a1, a2 being Real st ( (a1 * p1) + (a2 * p2) = 0.REAL 3 & a2 <> 0 ) ; ::_thesis: p1 p2 = 0.REAL 3 then consider a1, a2 being Real such that A7: ( a2 <> 0 & (a1 * p1) + (a2 * p2) = 0.REAL 3 ) ; A8: (1 / a2) * (a2 * p2) = (1 / a2) * (- (a1 * p1)) by A7, RVSUM_1:23 .= (1 / a2) * (((- 1) * a1) * p1) by RVSUM_1:49 .= (1 / a2) * ((- a1) * p1) ; A9: (1 / a2) * (a2 * p2) = (a2 * (1 / a2)) * p2 by EUCLID_4:4 .= 1 * p2 by A7, XCMPLX_1:106 ; A10: 1 * p2 = |[(1 * (p2 . 1)),(1 * (p2 . 2)),(1 * (p2 . 3))]| by Lm1 .= p2 by Th1 ; A11: 0.REAL 3 = |[0,0,0]| by FINSEQ_2:62; p1 p2 = p1 (((- a1) * (1 / a2)) * p1) by A8, A9, A10, EUCLID_4:4 .= p1 (((- a1) / a2) * p1) by XCMPLX_1:99 .= (((- a1) / a2) * p1) p1 by Lm14 .= ((- a1) / a2) * (p1 p1) by Lm14 .= ((- a1) / a2) * (0.REAL 3) by FINSEQ_2:62 .= |[(((- a1) / a2) * ((0.REAL 3) . 1)),(((- a1) / a2) * ((0.REAL 3) . 2)),(((- a1) / a2) * ((0.REAL 3) . 3))]| by Lm1 .= |[(((- a1) / a2) * 0),(((- a1) / a2) * ((0.REAL 3) . 2)),(((- a1) / a2) * ((0.REAL 3) . 3))]| by A11, FINSEQ_1:45 .= |[0,(((- a1) / a2) * 0),(((- a1) / a2) * ((0.REAL 3) . 3))]| by A11, FINSEQ_1:45 .= 0.REAL 3 by A11, FINSEQ_1:45 ; hence p1 p2 = 0.REAL 3 ; ::_thesis: verum end; end; end; hence p1 p2 = 0.REAL 3 ; ::_thesis: verum end; begin theorem :: EUCLID_8:4 |..| = 1 proof |(,)| = ((1 ^2) + (0 ^2)) + (0 ^2) by Lm12 .= 1 ; hence |..| = 1 by SQUARE_1:18; ::_thesis: verum end; theorem :: EUCLID_8:5 |..| = 1 proof |(,)| = ((0 ^2) + (1 ^2)) + (0 ^2) by Lm12 .= 1 ; hence |..| = 1 by SQUARE_1:18; ::_thesis: verum end; theorem :: EUCLID_8:6 |..| = 1 proof |(,)| = ((0 ^2) + (0 ^2)) + (1 ^2) by Lm12 .= 1 ; hence |..| = 1 by SQUARE_1:18; ::_thesis: verum end; theorem :: EUCLID_8:7 , are_orthogonal proof ( . 1 = 1 & . 2 = 0 & . 3 = 0 & . 1 = 0 & . 2 = 1 & . 3 = 0 ) by FINSEQ_1:45; then |(,)| = ((1 * 0) + (0 * 1)) + (0 * 0) by Lm5 .= 0 ; hence , are_orthogonal by RVSUM_1:def_17; ::_thesis: verum end; theorem :: EUCLID_8:8 , are_orthogonal proof ( . 1 = 1 & . 2 = 0 & . 3 = 0 & . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; then |(,)| = ((1 * 0) + (0 * 0)) + (0 * 1) by Lm5 .= 0 ; hence , are_orthogonal by RVSUM_1:def_17; ::_thesis: verum end; theorem :: EUCLID_8:9 , are_orthogonal proof ( . 1 = 0 & . 2 = 1 & . 3 = 0 & . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; then |(,)| = ((0 * 0) + (1 * 0)) + (0 * 1) by Lm5 .= 0 ; hence , are_orthogonal by RVSUM_1:def_17; ::_thesis: verum end; theorem :: EUCLID_8:10 |(,)| = 1 proof ( . 1 = 1 & . 2 = 0 & . 3 = 0 ) by FINSEQ_1:45; then |(,)| = ((1 * 1) + (0 * 0)) + (0 * 0) by Lm5 .= 1 ; hence |(,)| = 1 ; ::_thesis: verum end; theorem :: EUCLID_8:11 |(,)| = 1 proof ( . 1 = 0 & . 2 = 1 & . 3 = 0 ) by FINSEQ_1:45; then |(,)| = ((0 * 0) + (1 * 1)) + (0 * 0) by Lm5 .= 1 ; hence |(,)| = 1 ; ::_thesis: verum end; theorem :: EUCLID_8:12 |(,)| = 1 proof ( . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; then |(,)| = ((0 * 0) + (0 * 0)) + (1 * 1) by Lm5 .= 1 ; hence |(,)| = 1 ; ::_thesis: verum end; theorem Th13: :: EUCLID_8:13 for p being Element of REAL 3 holds |(p,|[0,0,0]|)| = 0 proof let p be Element of REAL 3; ::_thesis: |(p,|[0,0,0]|)| = 0 set e = |[0,0,0]|; ( |[0,0,0]| . 1 = 0 & |[0,0,0]| . 2 = 0 & |[0,0,0]| . 3 = 0 ) by FINSEQ_1:45; hence |(p,|[0,0,0]|)| = (((p . 1) * 0) + ((p . 2) * 0)) + ((p . 3) * 0) by Lm5 .= 0 ; ::_thesis: verum end; theorem :: EUCLID_8:14 canceled; theorem :: EUCLID_8:15 canceled; theorem :: EUCLID_8:16 = proof = |[((0 * 0) - (0 * 1)),((0 * 0) - (1 * 0)),((1 * 1) - (0 * 0))]| by Lm13 .= ; hence = ; ::_thesis: verum end; theorem :: EUCLID_8:17 = proof = |[((1 * 1) - (0 * 0)),((0 * 0) - (0 * 1)),((0 * 0) - (1 * 0))]| by Lm13 .= ; hence = ; ::_thesis: verum end; theorem :: EUCLID_8:18 = proof = |[((0 * 0) - (1 * 0)),((1 * 1) - (0 * 0)),((0 * 0) - (0 * 1))]| by Lm13 .= ; hence = ; ::_thesis: verum end; theorem :: EUCLID_8:19 = - proof A1: ( . 1 = 1 & . 2 = 0 & . 3 = 0 ) by FINSEQ_1:45; = |[((0 * 0) - (1 * 1)),((1 * 0) - (0 * 0)),((0 * 1) - (0 * 0))]| by Lm13 .= |[((- 1) * ( . 1)),((- 1) * ( . 2)),((- 1) * ( . 3))]| by A1 .= - by Lm1 ; hence = - ; ::_thesis: verum end; theorem :: EUCLID_8:20 = - proof A1: ( . 1 = 0 & . 2 = 1 & . 3 = 0 ) by FINSEQ_1:45; = |[((0 * 1) - (0 * 0)),((0 * 0) - (1 * 1)),((1 * 0) - (0 * 0))]| by Lm13 .= |[((- 1) * ( . 1)),((- 1) * ( . 2)),((- 1) * ( . 3))]| by A1 .= - by Lm1 ; hence = - ; ::_thesis: verum end; theorem :: EUCLID_8:21 = - proof A1: ( . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; = |[((1 * 0) - (0 * 0)),((0 * 1) - (0 * 0)),((0 * 0) - (1 * 1))]| by Lm13 .= |[((- 1) * ( . 1)),((- 1) * ( . 2)),((- 1) * ( . 3))]| by A1 .= - by Lm1 ; hence = - ; ::_thesis: verum end; theorem :: EUCLID_8:22 for p being Element of REAL 3 holds p |[0,0,0]| = |[0,0,0]| proof let p be Element of REAL 3; ::_thesis: p |[0,0,0]| = |[0,0,0]| p = |[(p . 1),(p . 2),(p . 3)]| by Th1; hence p |[0,0,0]| = |[(((p . 2) * 0) - ((p . 3) * 0)),(((p . 3) * 0) - ((p . 1) * 0)),(((p . 1) * 0) - ((p . 2) * 0))]| by Lm13 .= |[0,0,0]| ; ::_thesis: verum end; theorem :: EUCLID_8:23 canceled; theorem :: EUCLID_8:24 canceled; theorem Th25: :: EUCLID_8:25 for r being Element of REAL holds r * = |[r,0,0]| proof let r be Element of REAL ; ::_thesis: r * = |[r,0,0]| ( . 1 = 1 & . 2 = 0 & . 3 = 0 ) by FINSEQ_1:45; then r * = |[(r * 1),(r * 0),(r * 0)]| by Lm1 .= |[r,0,0]| ; hence r * = |[r,0,0]| ; ::_thesis: verum end; theorem Th26: :: EUCLID_8:26 for r being Element of REAL holds r * = |[0,r,0]| proof let r be Element of REAL ; ::_thesis: r * = |[0,r,0]| ( . 1 = 0 & . 2 = 1 & . 3 = 0 ) by FINSEQ_1:45; then r * = |[(r * 0),(r * 1),(r * 0)]| by Lm1 .= |[0,r,0]| ; hence r * = |[0,r,0]| ; ::_thesis: verum end; theorem Th27: :: EUCLID_8:27 for r being Element of REAL holds r * = |[0,0,r]| proof let r be Element of REAL ; ::_thesis: r * = |[0,0,r]| ( . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; then r * = |[(r * 0),(r * 0),(r * 1)]| by Lm1 .= |[0,0,r]| ; hence r * = |[0,0,r]| ; ::_thesis: verum end; theorem :: EUCLID_8:28 for p being Element of REAL 3 holds 1 * p = p by RFUNCT_1:21; theorem :: EUCLID_8:29 canceled; theorem :: EUCLID_8:30 canceled; theorem :: EUCLID_8:31 - = |[(- 1),0,0]| proof A1: ( . 1 = 1 & . 2 = 0 & . 3 = 0 ) by FINSEQ_1:45; - = |[((- 1) * ( . 1)),((- 1) * ( . 2)),((- 1) * ( . 3))]| by Lm1 .= |[(- 1),0,0]| by A1 ; hence - = |[(- 1),0,0]| ; ::_thesis: verum end; theorem :: EUCLID_8:32 - = |[0,(- 1),0]| proof A1: ( . 1 = 0 & . 2 = 1 & . 3 = 0 ) by FINSEQ_1:45; - = |[((- 1) * ( . 1)),((- 1) * ( . 2)),((- 1) * ( . 3))]| by Lm1 .= |[0,(- 1),0]| by A1 ; hence - = |[0,(- 1),0]| ; ::_thesis: verum end; theorem :: EUCLID_8:33 - = |[0,0,(- 1)]| proof A1: ( . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; - = |[((- 1) * ( . 1)),((- 1) * ( . 2)),((- 1) * ( . 3))]| by Lm1 .= |[0,0,(- 1)]| by A1 ; hence - = |[0,0,(- 1)]| ; ::_thesis: verum end; theorem :: EUCLID_8:34 for p being Element of REAL 3 holds 0 * p = |[0,0,0]| proof let p be Element of REAL 3; ::_thesis: 0 * p = |[0,0,0]| thus 0 * p = |[(0 * (p . 1)),(0 * (p . 2)),(0 * (p . 3))]| by Lm1 .= |[0,0,0]| ; ::_thesis: verum end; theorem :: EUCLID_8:35 canceled; theorem :: EUCLID_8:36 canceled; theorem :: EUCLID_8:37 for p being Element of REAL 3 holds p = (((p . 1) * ) + ((p . 2) * )) + ((p . 3) * ) proof let p be Element of REAL 3; ::_thesis: p = (((p . 1) * ) + ((p . 2) * )) + ((p . 3) * ) A1: ( . 1 = 1 & . 2 = 0 & . 3 = 0 ) by FINSEQ_1:45; A2: ( . 1 = 0 & . 2 = 1 & . 3 = 0 ) by FINSEQ_1:45; A3: ( . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; A4: (((p . 1) * ) + ((p . 2) * )) + ((p . 3) * ) = (|[((p . 1) * 1),((p . 1) * 0),((p . 1) * 0)]| + ((p . 2) * )) + ((p . 3) * ) by A1, Lm1 .= (|[(p . 1),0,0]| + |[((p . 2) * 0),((p . 2) * 1),((p . 2) * 0)]|) + ((p . 3) * ) by A2, Lm1 .= (|[(p . 1),0,0]| + |[0,(p . 2),0]|) + |[((p . 3) * 0),((p . 3) * 0),((p . 3) * 1)]| by A3, Lm1 .= (|[(p . 1),0,0]| + |[0,(p . 2),0]|) + |[0,0,(p . 3)]| ; A5: ( |[(p . 1),0,0]| . 1 = p . 1 & |[(p . 1),0,0]| . 2 = 0 & |[(p . 1),0,0]| . 3 = 0 ) by FINSEQ_1:45; A6: ( |[0,(p . 2),0]| . 1 = 0 & |[0,(p . 2),0]| . 2 = p . 2 & |[0,(p . 2),0]| . 3 = 0 ) by FINSEQ_1:45; A7: ( |[0,0,(p . 3)]| . 1 = 0 & |[0,0,(p . 3)]| . 2 = 0 & |[0,0,(p . 3)]| . 3 = p . 3 ) by FINSEQ_1:45; A8: (((p . 1) * ) + ((p . 2) * )) + ((p . 3) * ) = |[((p . 1) + 0),(0 + (p . 2)),(0 + 0)]| + |[0,0,(p . 3)]| by A4, A5, A6, Lm2 .= |[(p . 1),(p . 2),0]| + |[0,0,(p . 3)]| ; ( |[(p . 1),(p . 2),0]| . 1 = p . 1 & |[(p . 1),(p . 2),0]| . 2 = p . 2 & |[(p . 1),(p . 2),0]| . 3 = 0 ) by FINSEQ_1:45; then (((p . 1) * ) + ((p . 2) * )) + ((p . 3) * ) = |[((p . 1) + 0),((p . 2) + 0),(0 + (p . 3))]| by A7, A8, Lm2 .= |[(p . 1),(p . 2),(p . 3)]| ; hence p = (((p . 1) * ) + ((p . 2) * )) + ((p . 3) * ) by Th1; ::_thesis: verum end; theorem Th38: :: EUCLID_8:38 for r being Element of REAL for p being Element of REAL 3 holds r * p = (((r * (p . 1)) * ) + ((r * (p . 2)) * )) + ((r * (p . 3)) * ) proof let r be Element of REAL ; ::_thesis: for p being Element of REAL 3 holds r * p = (((r * (p . 1)) * ) + ((r * (p . 2)) * )) + ((r * (p . 3)) * ) let p be Element of REAL 3; ::_thesis: r * p = (((r * (p . 1)) * ) + ((r * (p . 2)) * )) + ((r * (p . 3)) * ) A1: ( . 1 = 1 & . 2 = 0 & . 3 = 0 ) by FINSEQ_1:45; A2: ( . 1 = 0 & . 2 = 1 & . 3 = 0 ) by FINSEQ_1:45; A3: ( . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; A4: (((r * (p . 1)) * ) + ((r * (p . 2)) * )) + ((r * (p . 3)) * ) = (|[((r * (p . 1)) * 1),((r * (p . 1)) * 0),((r * (p . 1)) * 0)]| + ((r * (p . 2)) * )) + ((r * (p . 3)) * ) by A1, Lm1 .= (|[(r * (p . 1)),0,0]| + |[((r * (p . 2)) * 0),((r * (p . 2)) * 1),((r * (p . 2)) * 0)]|) + ((r * (p . 3)) * ) by A2, Lm1 .= (|[(r * (p . 1)),0,0]| + |[0,(r * (p . 2)),0]|) + |[((r * (p . 3)) * 0),((r * (p . 3)) * 0),((r * (p . 3)) * 1)]| by A3, Lm1 .= (|[(r * (p . 1)),0,0]| + |[0,(r * (p . 2)),0]|) + |[0,0,(r * (p . 3))]| ; A5: ( |[(r * (p . 1)),0,0]| . 1 = r * (p . 1) & |[(r * (p . 1)),0,0]| . 2 = 0 & |[(r * (p . 1)),0,0]| . 3 = 0 ) by FINSEQ_1:45; A6: ( |[0,(r * (p . 2)),0]| . 1 = 0 & |[0,(r * (p . 2)),0]| . 2 = r * (p . 2) & |[0,(r * (p . 2)),0]| . 3 = 0 ) by FINSEQ_1:45; A7: ( |[0,0,(r * (p . 3))]| . 1 = 0 & |[0,0,(r * (p . 3))]| . 2 = 0 & |[0,0,(r * (p . 3))]| . 3 = r * (p . 3) ) by FINSEQ_1:45; A8: (((r * (p . 1)) * ) + ((r * (p . 2)) * )) + ((r * (p . 3)) * ) = |[((r * (p . 1)) + 0),(0 + (r * (p . 2))),(0 + 0)]| + |[0,0,(r * (p . 3))]| by A4, A5, A6, Lm2 .= |[(r * (p . 1)),(r * (p . 2)),0]| + |[0,0,(r * (p . 3))]| ; ( |[(r * (p . 1)),(r * (p . 2)),0]| . 1 = r * (p . 1) & |[(r * (p . 1)),(r * (p . 2)),0]| . 2 = r * (p . 2) & |[(r * (p . 1)),(r * (p . 2)),0]| . 3 = 0 ) by FINSEQ_1:45; then (((r * (p . 1)) * ) + ((r * (p . 2)) * )) + ((r * (p . 3)) * ) = |[((r * (p . 1)) + 0),((r * (p . 2)) + 0),(0 + (r * (p . 3)))]| by A7, A8, Lm2 .= |[(r * (p . 1)),(r * (p . 2)),(r * (p . 3))]| ; hence r * p = (((r * (p . 1)) * ) + ((r * (p . 2)) * )) + ((r * (p . 3)) * ) by Lm1; ::_thesis: verum end; theorem Th39: :: EUCLID_8:39 for x, y, z being Element of REAL holds |[x,y,z]| = ((x * ) + (y * )) + (z * ) proof let x, y, z be Element of REAL ; ::_thesis: |[x,y,z]| = ((x * ) + (y * )) + (z * ) A1: ( . 1 = 1 & . 2 = 0 & . 3 = 0 ) by FINSEQ_1:45; A2: ( . 1 = 0 & . 2 = 1 & . 3 = 0 ) by FINSEQ_1:45; A3: ( . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; set p = |[x,y,z]|; A4: ((x * ) + (y * )) + (z * ) = (|[(x * 1),(x * 0),(x * 0)]| + (y * )) + (z * ) by A1, Lm1 .= (|[x,0,0]| + |[(y * 0),(y * 1),(y * 0)]|) + (z * ) by A2, Lm1 .= (|[x,0,0]| + |[0,y,0]|) + |[(z * 0),(z * 0),(z * 1)]| by A3, Lm1 .= (|[x,0,0]| + |[0,y,0]|) + |[0,0,z]| ; A5: ( |[x,0,0]| . 1 = x & |[x,0,0]| . 2 = 0 & |[x,0,0]| . 3 = 0 ) by FINSEQ_1:45; A6: ( |[0,y,0]| . 1 = 0 & |[0,y,0]| . 2 = y & |[0,y,0]| . 3 = 0 ) by FINSEQ_1:45; A7: ( |[0,0,z]| . 1 = 0 & |[0,0,z]| . 2 = 0 & |[0,0,z]| . 3 = z ) by FINSEQ_1:45; A8: ((x * ) + (y * )) + (z * ) = |[(x + 0),(y + 0),(0 + 0)]| + |[0,0,z]| by A4, A5, A6, Lm2 .= |[x,y,0]| + |[0,0,z]| ; ( |[x,y,0]| . 1 = x & |[x,y,0]| . 2 = y & |[x,y,0]| . 3 = 0 ) by FINSEQ_1:45; then ((x * ) + (y * )) + (z * ) = |[(x + 0),(y + 0),(0 + z)]| by A7, A8, Lm2 .= |[x,y,z]| ; hence |[x,y,z]| = ((x * ) + (y * )) + (z * ) ; ::_thesis: verum end; theorem Th40: :: EUCLID_8:40 for r, x, y, z being Element of REAL holds r * |[x,y,z]| = (((r * x) * ) + ((r * y) * )) + ((r * z) * ) proof let r, x, y, z be Element of REAL ; ::_thesis: r * |[x,y,z]| = (((r * x) * ) + ((r * y) * )) + ((r * z) * ) A1: ( . 1 = 1 & . 2 = 0 & . 3 = 0 ) by FINSEQ_1:45; A2: ( . 1 = 0 & . 2 = 1 & . 3 = 0 ) by FINSEQ_1:45; A3: ( . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; set p = |[x,y,z]|; A4: (((r * x) * ) + ((r * y) * )) + ((r * z) * ) = (|[((r * x) * 1),((r * x) * 0),((r * x) * 0)]| + ((r * y) * )) + ((r * z) * ) by A1, Lm1 .= (|[(r * x),0,0]| + |[((r * y) * 0),((r * y) * 1),((r * y) * 0)]|) + ((r * z) * ) by A2, Lm1 .= (|[(r * x),0,0]| + |[0,(r * y),0]|) + |[((r * z) * 0),((r * z) * 0),((r * z) * 1)]| by A3, Lm1 .= (|[(r * x),0,0]| + |[0,(r * y),0]|) + |[0,0,(r * z)]| ; A5: ( |[(r * x),0,0]| . 1 = r * x & |[(r * x),0,0]| . 2 = 0 & |[(r * x),0,0]| . 3 = 0 ) by FINSEQ_1:45; A6: ( |[0,(r * y),0]| . 1 = 0 & |[0,(r * y),0]| . 2 = r * y & |[0,(r * y),0]| . 3 = 0 ) by FINSEQ_1:45; A7: ( |[0,0,(r * z)]| . 1 = 0 & |[0,0,(r * z)]| . 2 = 0 & |[0,0,(r * z)]| . 3 = r * z ) by FINSEQ_1:45; A8: (((r * x) * ) + ((r * y) * )) + ((r * z) * ) = |[((r * x) + 0),((r * y) + 0),(0 + 0)]| + |[0,0,(r * z)]| by A4, A5, A6, Lm2 .= |[(r * x),(r * y),0]| + |[0,0,(r * z)]| ; ( |[(r * x),(r * y),0]| . 1 = r * x & |[(r * x),(r * y),0]| . 2 = r * y & |[(r * x),(r * y),0]| . 3 = 0 ) by FINSEQ_1:45; then (((r * x) * ) + ((r * y) * )) + ((r * z) * ) = |[((r * x) + 0),((r * y) + 0),(0 + (r * z))]| by A7, A8, Lm2 .= |[(r * x),(r * y),(r * z)]| ; hence r * |[x,y,z]| = (((r * x) * ) + ((r * y) * )) + ((r * z) * ) by Lm6; ::_thesis: verum end; theorem :: EUCLID_8:41 for p being Element of REAL 3 holds - p = ((- ((p . 1) * )) - ((p . 2) * )) - ((p . 3) * ) proof let p be Element of REAL 3; ::_thesis: - p = ((- ((p . 1) * )) - ((p . 2) * )) - ((p . 3) * ) - 1 is Element of REAL by XREAL_0:def_1; then - p = ((((- 1) * (p . 1)) * ) + (((- 1) * (p . 2)) * )) + (((- 1) * (p . 3)) * ) by Th38 .= ((- ((p . 1) * )) + (((- 1) * (p . 2)) * )) + ((- (p . 3)) * ) by RVSUM_1:49 .= ((- ((p . 1) * )) + (- ((p . 2) * ))) + (((- 1) * (p . 3)) * ) by RVSUM_1:49 .= ((- ((p . 1) * )) - ((p . 2) * )) - ((p . 3) * ) by RVSUM_1:49 ; hence - p = ((- ((p . 1) * )) - ((p . 2) * )) - ((p . 3) * ) ; ::_thesis: verum end; theorem :: EUCLID_8:42 for x, y, z being Element of REAL holds - |[x,y,z]| = ((- (x * )) - (y * )) - (z * ) proof let x, y, z be Element of REAL ; ::_thesis: - |[x,y,z]| = ((- (x * )) - (y * )) - (z * ) - 1 in REAL by XREAL_0:def_1; then - |[x,y,z]| = ((((- 1) * x) * ) + (((- 1) * y) * )) + (((- 1) * z) * ) by Th40 .= ((- (x * )) + (((- 1) * y) * )) + ((- z) * ) by RVSUM_1:49 .= ((- (x * )) + (- (y * ))) + (((- 1) * z) * ) by RVSUM_1:49 .= ((- (x * )) - (y * )) - (z * ) by RVSUM_1:49 ; hence - |[x,y,z]| = ((- (x * )) - (y * )) - (z * ) ; ::_thesis: verum end; theorem :: EUCLID_8:43 for p1, p2 being Element of REAL 3 holds p1 + p2 = ((((p1 . 1) + (p2 . 1)) * ) + (((p1 . 2) + (p2 . 2)) * )) + (((p1 . 3) + (p2 . 3)) * ) proof let p1, p2 be Element of REAL 3; ::_thesis: p1 + p2 = ((((p1 . 1) + (p2 . 1)) * ) + (((p1 . 2) + (p2 . 2)) * )) + (((p1 . 3) + (p2 . 3)) * ) A1: ( . 1 = 1 & . 2 = 0 & . 3 = 0 ) by FINSEQ_1:45; A2: ( . 1 = 0 & . 2 = 1 & . 3 = 0 ) by FINSEQ_1:45; A3: ( . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; A4: (p1 + p2) . 1 = (p1 . 1) + (p2 . 1) by RVSUM_1:11; A5: (p1 + p2) . 2 = (p1 . 2) + (p2 . 2) by RVSUM_1:11; A6: (p1 + p2) . 3 = (p1 . 3) + (p2 . 3) by RVSUM_1:11; A7: ((((p1 + p2) . 1) * ) + (((p1 + p2) . 2) * )) + (((p1 + p2) . 3) * ) = (|[(((p1 + p2) . 1) * 1),(((p1 + p2) . 1) * 0),(((p1 + p2) . 1) * 0)]| + (((p1 + p2) . 2) * )) + (((p1 + p2) . 3) * ) by A1, Lm1 .= (|[((p1 + p2) . 1),0,0]| + |[(((p1 + p2) . 2) * 0),(((p1 + p2) . 2) * 1),(((p1 + p2) . 2) * 0)]|) + (((p1 + p2) . 3) * ) by A2, Lm1 .= (|[((p1 + p2) . 1),0,0]| + |[0,((p1 + p2) . 2),0]|) + |[(((p1 + p2) . 3) * 0),(((p1 + p2) . 3) * 0),(((p1 + p2) . 3) * 1)]| by A3, Lm1 .= (|[((p1 + p2) . 1),0,0]| + |[0,((p1 + p2) . 2),0]|) + |[0,0,((p1 + p2) . 3)]| ; A8: ( |[((p1 + p2) . 1),0,0]| . 1 = (p1 + p2) . 1 & |[((p1 + p2) . 1),0,0]| . 2 = 0 & |[((p1 + p2) . 1),0,0]| . 3 = 0 ) by FINSEQ_1:45; A9: ( |[0,((p1 + p2) . 2),0]| . 1 = 0 & |[0,((p1 + p2) . 2),0]| . 2 = (p1 + p2) . 2 & |[0,((p1 + p2) . 2),0]| . 3 = 0 ) by FINSEQ_1:45; A10: ( |[0,0,((p1 + p2) . 3)]| . 1 = 0 & |[0,0,((p1 + p2) . 3)]| . 2 = 0 & |[0,0,((p1 + p2) . 3)]| . 3 = (p1 + p2) . 3 ) by FINSEQ_1:45; A11: ((((p1 + p2) . 1) * ) + (((p1 + p2) . 2) * )) + (((p1 + p2) . 3) * ) = |[(((p1 + p2) . 1) + 0),(0 + ((p1 + p2) . 2)),(0 + 0)]| + |[0,0,((p1 + p2) . 3)]| by A7, A8, A9, Lm2 .= |[((p1 + p2) . 1),((p1 + p2) . 2),0]| + |[0,0,((p1 + p2) . 3)]| ; ( |[((p1 + p2) . 1),((p1 + p2) . 2),0]| . 1 = (p1 + p2) . 1 & |[((p1 + p2) . 1),((p1 + p2) . 2),0]| . 2 = (p1 + p2) . 2 & |[((p1 + p2) . 1),((p1 + p2) . 2),0]| . 3 = 0 ) by FINSEQ_1:45; then ((((p1 + p2) . 1) * ) + (((p1 + p2) . 2) * )) + (((p1 + p2) . 3) * ) = |[(((p1 + p2) . 1) + 0),(((p1 + p2) . 2) + 0),(0 + ((p1 + p2) . 3))]| by A10, A11, Lm2 .= |[((p1 + p2) . 1),((p1 + p2) . 2),((p1 + p2) . 3)]| ; hence p1 + p2 = ((((p1 . 1) + (p2 . 1)) * ) + (((p1 . 2) + (p2 . 2)) * )) + (((p1 . 3) + (p2 . 3)) * ) by A4, A5, A6, Th1; ::_thesis: verum end; theorem Th44: :: EUCLID_8:44 for p1, p2 being Element of REAL 3 holds p1 - p2 = ((((p1 . 1) - (p2 . 1)) * ) + (((p1 . 2) - (p2 . 2)) * )) + (((p1 . 3) - (p2 . 3)) * ) proof let p1, p2 be Element of REAL 3; ::_thesis: p1 - p2 = ((((p1 . 1) - (p2 . 1)) * ) + (((p1 . 2) - (p2 . 2)) * )) + (((p1 . 3) - (p2 . 3)) * ) A1: ( . 1 = 1 & . 2 = 0 & . 3 = 0 ) by FINSEQ_1:45; A2: ( . 1 = 0 & . 2 = 1 & . 3 = 0 ) by FINSEQ_1:45; A3: ( . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; A4: (p1 - p2) . 1 = (p1 . 1) - (p2 . 1) by RVSUM_1:27; A5: (p1 - p2) . 2 = (p1 . 2) - (p2 . 2) by RVSUM_1:27; A6: (p1 - p2) . 3 = (p1 . 3) - (p2 . 3) by RVSUM_1:27; A7: ((((p1 - p2) . 1) * ) + (((p1 - p2) . 2) * )) + (((p1 - p2) . 3) * ) = (|[(((p1 - p2) . 1) * 1),(((p1 - p2) . 1) * 0),(((p1 - p2) . 1) * 0)]| + (((p1 - p2) . 2) * )) + (((p1 - p2) . 3) * ) by A1, Lm1 .= (|[((p1 - p2) . 1),0,0]| + |[(((p1 - p2) . 2) * 0),(((p1 - p2) . 2) * 1),(((p1 - p2) . 2) * 0)]|) + (((p1 - p2) . 3) * ) by A2, Lm1 .= (|[((p1 - p2) . 1),0,0]| + |[0,((p1 - p2) . 2),0]|) + |[(((p1 - p2) . 3) * 0),(((p1 - p2) . 3) * 0),(((p1 - p2) . 3) * 1)]| by A3, Lm1 .= (|[((p1 - p2) . 1),0,0]| + |[0,((p1 - p2) . 2),0]|) + |[0,0,((p1 - p2) . 3)]| ; A8: ( |[((p1 - p2) . 1),0,0]| . 1 = (p1 - p2) . 1 & |[((p1 - p2) . 1),0,0]| . 2 = 0 & |[((p1 - p2) . 1),0,0]| . 3 = 0 ) by FINSEQ_1:45; A9: ( |[0,((p1 - p2) . 2),0]| . 1 = 0 & |[0,((p1 - p2) . 2),0]| . 2 = (p1 - p2) . 2 & |[0,((p1 - p2) . 2),0]| . 3 = 0 ) by FINSEQ_1:45; A10: ( |[0,0,((p1 - p2) . 3)]| . 1 = 0 & |[0,0,((p1 - p2) . 3)]| . 2 = 0 & |[0,0,((p1 - p2) . 3)]| . 3 = (p1 - p2) . 3 ) by FINSEQ_1:45; A11: ((((p1 - p2) . 1) * ) + (((p1 - p2) . 2) * )) + (((p1 - p2) . 3) * ) = |[(((p1 - p2) . 1) + 0),(0 + ((p1 - p2) . 2)),(0 + 0)]| + |[0,0,((p1 - p2) . 3)]| by A7, A8, A9, Lm2 .= |[((p1 - p2) . 1),((p1 - p2) . 2),0]| + |[0,0,((p1 - p2) . 3)]| ; ( |[((p1 - p2) . 1),((p1 - p2) . 2),0]| . 1 = (p1 - p2) . 1 & |[((p1 - p2) . 1),((p1 - p2) . 2),0]| . 2 = (p1 - p2) . 2 & |[((p1 - p2) . 1),((p1 - p2) . 2),0]| . 3 = 0 ) by FINSEQ_1:45; then ((((p1 - p2) . 1) * ) + (((p1 - p2) . 2) * )) + (((p1 - p2) . 3) * ) = |[(((p1 - p2) . 1) + 0),(((p1 - p2) . 2) + 0),(0 + ((p1 - p2) . 3))]| by A10, A11, Lm2 .= |[((p1 - p2) . 1),((p1 - p2) . 2),((p1 - p2) . 3)]| ; hence p1 - p2 = ((((p1 . 1) - (p2 . 1)) * ) + (((p1 . 2) - (p2 . 2)) * )) + (((p1 . 3) - (p2 . 3)) * ) by A4, A5, A6, Th1; ::_thesis: verum end; theorem :: EUCLID_8:45 for x1, x2, x3, y1, y2, y3 being Element of REAL holds |[x1,x2,x3]| + |[y1,y2,y3]| = (((x1 + y1) * ) + ((x2 + y2) * )) + ((x3 + y3) * ) proof let x1, x2, x3, y1, y2, y3 be Element of REAL ; ::_thesis: |[x1,x2,x3]| + |[y1,y2,y3]| = (((x1 + y1) * ) + ((x2 + y2) * )) + ((x3 + y3) * ) A1: ( . 1 = 1 & . 2 = 0 & . 3 = 0 ) by FINSEQ_1:45; A2: ( . 1 = 0 & . 2 = 1 & . 3 = 0 ) by FINSEQ_1:45; A3: ( . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; A4: (((x1 + y1) * ) + ((x2 + y2) * )) + ((x3 + y3) * ) = (|[((x1 + y1) * 1),((x1 + y1) * 0),((x1 + y1) * 0)]| + ((x2 + y2) * )) + ((x3 + y3) * ) by A1, Lm1 .= (|[(x1 + y1),0,0]| + |[((x2 + y2) * 0),((x2 + y2) * 1),((x2 + y2) * 0)]|) + ((x3 + y3) * ) by A2, Lm1 .= (|[(x1 + y1),0,0]| + |[0,(x2 + y2),0]|) + |[((x3 + y3) * 0),((x3 + y3) * 0),((x3 + y3) * 1)]| by A3, Lm1 .= (|[(x1 + y1),0,0]| + |[0,(x2 + y2),0]|) + |[0,0,(x3 + y3)]| ; A5: ( |[(x1 + y1),0,0]| . 1 = x1 + y1 & |[(x1 + y1),0,0]| . 2 = 0 & |[(x1 + y1),0,0]| . 3 = 0 ) by FINSEQ_1:45; A6: ( |[0,(x2 + y2),0]| . 1 = 0 & |[0,(x2 + y2),0]| . 2 = x2 + y2 & |[0,(x2 + y2),0]| . 3 = 0 ) by FINSEQ_1:45; A7: ( |[0,0,(x3 + y3)]| . 1 = 0 & |[0,0,(x3 + y3)]| . 2 = 0 & |[0,0,(x3 + y3)]| . 3 = x3 + y3 ) by FINSEQ_1:45; A8: (((x1 + y1) * ) + ((x2 + y2) * )) + ((x3 + y3) * ) = |[((x1 + y1) + 0),(0 + (x2 + y2)),(0 + 0)]| + |[0,0,(x3 + y3)]| by A4, A5, A6, Lm2 .= |[(x1 + y1),(x2 + y2),0]| + |[0,0,(x3 + y3)]| ; ( |[(x1 + y1),(x2 + y2),0]| . 1 = x1 + y1 & |[(x1 + y1),(x2 + y2),0]| . 2 = x2 + y2 & |[(x1 + y1),(x2 + y2),0]| . 3 = 0 ) by FINSEQ_1:45; then A9: (((x1 + y1) * ) + ((x2 + y2) * )) + ((x3 + y3) * ) = |[((x1 + y1) + 0),((x2 + y2) + 0),(0 + (x3 + y3))]| by A8, A7, Lm2 .= |[(x1 + y1),(x2 + y2),(x3 + y3)]| ; A10: ( |[x1,x2,x3]| . 1 = x1 & |[x1,x2,x3]| . 2 = x2 & |[x1,x2,x3]| . 3 = x3 ) by FINSEQ_1:45; ( |[y1,y2,y3]| . 1 = y1 & |[y1,y2,y3]| . 2 = y2 & |[y1,y2,y3]| . 3 = y3 ) by FINSEQ_1:45; hence |[x1,x2,x3]| + |[y1,y2,y3]| = (((x1 + y1) * ) + ((x2 + y2) * )) + ((x3 + y3) * ) by A10, Lm2, A9; ::_thesis: verum end; theorem :: EUCLID_8:46 for x1, x2, x3, y1, y2, y3 being Element of REAL holds |[x1,x2,x3]| - |[y1,y2,y3]| = (((x1 - y1) * ) + ((x2 - y2) * )) + ((x3 - y3) * ) proof let x1, x2, x3, y1, y2, y3 be Element of REAL ; ::_thesis: |[x1,x2,x3]| - |[y1,y2,y3]| = (((x1 - y1) * ) + ((x2 - y2) * )) + ((x3 - y3) * ) A1: |[y1,y2,y3]| . 1 = y1 by FINSEQ_1:45; A2: |[y1,y2,y3]| . 2 = y2 by FINSEQ_1:45; A3: |[y1,y2,y3]| . 3 = y3 by FINSEQ_1:45; A4: (|[x1,x2,x3]| . 1) - (|[y1,y2,y3]| . 1) = x1 - y1 by A1, FINSEQ_1:45; A5: (|[x1,x2,x3]| . 2) - (|[y1,y2,y3]| . 2) = x2 - y2 by A2, FINSEQ_1:45; (|[x1,x2,x3]| . 3) - (|[y1,y2,y3]| . 3) = x3 - y3 by A3, FINSEQ_1:45; hence |[x1,x2,x3]| - |[y1,y2,y3]| = (((x1 - y1) * ) + ((x2 - y2) * )) + ((x3 - y3) * ) by A4, A5, Th44; ::_thesis: verum end; theorem :: EUCLID_8:47 for p1, p2, p3 being Element of REAL 3 holds ( (((p1 . 1) * ) + ((p1 . 2) * )) + ((p1 . 3) * ) = ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) iff (((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * ) = ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) ) proof let p1, p2, p3 be Element of REAL 3; ::_thesis: ( (((p1 . 1) * ) + ((p1 . 2) * )) + ((p1 . 3) * ) = ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) iff (((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * ) = ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) ) A1: for r1, r2 being Element of REAL for R being Element of 3 -tuples_on REAL holds (r1 * R) - (r2 * R) = (r1 - r2) * R proof let r1, r2 be Element of REAL ; ::_thesis: for R being Element of 3 -tuples_on REAL holds (r1 * R) - (r2 * R) = (r1 - r2) * R let R be Element of 3 -tuples_on REAL; ::_thesis: (r1 * R) - (r2 * R) = (r1 - r2) * R (r1 * R) - (r2 * R) = (r1 * R) + (((- 1) * r2) * R) by RVSUM_1:49 .= (r1 + (- r2)) * R by RVSUM_1:50 ; hence (r1 * R) - (r2 * R) = (r1 - r2) * R ; ::_thesis: verum end; A2: ( . 1 = 1 & . 2 = 0 & . 3 = 0 ) by FINSEQ_1:45; A3: ( . 1 = 0 & . 2 = 1 & . 3 = 0 ) by FINSEQ_1:45; A4: ( . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; thus ( (((p1 . 1) * ) + ((p1 . 2) * )) + ((p1 . 3) * ) = ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) implies (((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * ) = ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) ) ::_thesis: ( (((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * ) = ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) implies (((p1 . 1) * ) + ((p1 . 2) * )) + ((p1 . 3) * ) = ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) ) proof assume (((p1 . 1) * ) + ((p1 . 2) * )) + ((p1 . 3) * ) = ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) ; ::_thesis: (((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * ) = ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) then (((((p1 . 1) * ) + (((p1 . 2) * ) + ((p1 . 3) * ))) - ((p3 . 1) * )) - ((p3 . 2) * )) - ((p3 . 3) * ) = (((((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 1) * )) - ((p3 . 2) * )) - ((p3 . 3) * ) by RVSUM_1:15; then (((((p1 . 1) * ) - ((p3 . 1) * )) + (((p1 . 2) * ) + ((p1 . 3) * ))) - ((p3 . 2) * )) - ((p3 . 3) * ) = (((((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 1) * )) - ((p3 . 2) * )) - ((p3 . 3) * ) by RVSUM_1:15; then (((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) * ) + ((p1 . 3) * ))) - ((p3 . 2) * )) - ((p3 . 3) * ) = (((((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 1) * )) - ((p3 . 2) * )) - ((p3 . 3) * ) by A1; then ((((((p1 . 1) - (p3 . 1)) * ) + ((p1 . 2) * )) + ((p1 . 3) * )) - ((p3 . 2) * )) - ((p3 . 3) * ) = (((((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 1) * )) - ((p3 . 2) * )) - ((p3 . 3) * ) by RVSUM_1:15; then ((((((p1 . 1) - (p3 . 1)) * ) + ((p1 . 2) * )) - ((p3 . 2) * )) + ((p1 . 3) * )) - ((p3 . 3) * ) = (((((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 1) * )) - ((p3 . 2) * )) - ((p3 . 3) * ) by RVSUM_1:15; then (((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) * ) + (- ((p3 . 2) * )))) + ((p1 . 3) * )) + (- ((p3 . 3) * )) = (((((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 1) * )) - ((p3 . 2) * )) - ((p3 . 3) * ) by RVSUM_1:15; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) * ) - ((p3 . 2) * ))) + (((p1 . 3) * ) - ((p3 . 3) * )) = (((((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 1) * )) - ((p3 . 2) * )) - ((p3 . 3) * ) by RVSUM_1:15; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) * ) - ((p3 . 3) * )) = (((((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 1) * )) - ((p3 . 2) * )) - ((p3 . 3) * ) by A1; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = (((((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 1) * )) - ((p3 . 2) * )) - ((p3 . 3) * ) by A1; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = (((((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) - ((p3 . 1) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 2) * )) - ((p3 . 3) * ) by RVSUM_1:15; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = (((((((p2 . 1) + (p3 . 1)) * ) - ((p3 . 1) * )) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 2) * )) - ((p3 . 3) * ) by RVSUM_1:15; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = (((((((p2 . 1) + (p3 . 1)) * ) - ((p3 . 1) * )) + (((p2 . 2) + (p3 . 2)) * )) - ((p3 . 2) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 3) * ) by RVSUM_1:15; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = (((((((p2 . 1) * ) + ((p3 . 1) * )) + (- ((p3 . 1) * ))) + (((p2 . 2) + (p3 . 2)) * )) - ((p3 . 2) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 3) * ) by RVSUM_1:50; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = ((((((p2 . 1) * ) + (((p3 . 1) * ) - ((p3 . 1) * ))) + (((p2 . 2) + (p3 . 2)) * )) - ((p3 . 2) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 3) * ) by RVSUM_1:15; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = ((((((p2 . 1) * ) + (0.REAL 3)) + (((p2 . 2) + (p3 . 2)) * )) - ((p3 . 2) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 3) * ) by EUCLIDLP:2; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = (((((p2 . 1) * ) + (((p2 . 2) + (p3 . 2)) * )) - ((p3 . 2) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 3) * ) by EUCLID_4:1; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = (((((p2 . 1) * ) + (((p2 . 2) * ) + ((p3 . 2) * ))) - ((p3 . 2) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 3) * ) by RVSUM_1:50; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = ((((((p2 . 1) * ) + ((p2 . 2) * )) + ((p3 . 2) * )) + (- ((p3 . 2) * ))) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 3) * ) by RVSUM_1:15; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = (((((p2 . 1) * ) + ((p2 . 2) * )) + (((p3 . 2) * ) + (- ((p3 . 2) * )))) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 3) * ) by RVSUM_1:15; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = (((((p2 . 1) * ) + ((p2 . 2) * )) + (0.REAL 3)) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 3) * ) by EUCLIDLP:2; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = ((((p2 . 1) * ) + ((p2 . 2) * )) + (((p2 . 3) + (p3 . 3)) * )) - ((p3 . 3) * ) by EUCLID_4:1; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = ((((p2 . 1) * ) + ((p2 . 2) * )) + (((p2 . 3) * ) + ((p3 . 3) * ))) - ((p3 . 3) * ) by RVSUM_1:50; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = (((((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * )) + ((p3 . 3) * )) - ((p3 . 3) * ) by RVSUM_1:15; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = ((((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * )) + (((p3 . 3) * ) + (- ((p3 . 3) * ))) by RVSUM_1:15; then ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) = ((((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * )) + (0.REAL 3) by EUCLIDLP:2; hence (((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * ) = ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) by EUCLID_4:1; ::_thesis: verum end; now__::_thesis:_(_(((p2_._1)_*_)_+_((p2_._2)_*_))_+_((p2_._3)_*_)_=_((((p1_._1)_-_(p3_._1))_*_)_+_(((p1_._2)_-_(p3_._2))_*_))_+_(((p1_._3)_-_(p3_._3))_*_)_&_(((p2_._1)_*_)_+_((p2_._2)_*_))_+_((p2_._3)_*_)_=_((((p1_._1)_-_(p3_._1))_*_)_+_(((p1_._2)_-_(p3_._2))_*_))_+_(((p1_._3)_-_(p3_._3))_*_)_implies_(((p1_._1)_*_)_+_((p1_._2)_*_))_+_((p1_._3)_*_)_=_((((p2_._1)_+_(p3_._1))_*_)_+_(((p2_._2)_+_(p3_._2))_*_))_+_(((p2_._3)_+_(p3_._3))_*_)_) assume A5: (((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * ) = ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) ; ::_thesis: ( (((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * ) = ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) implies (((p1 . 1) * ) + ((p1 . 2) * )) + ((p1 . 3) * ) = ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) ) ((((((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * )) + ((p3 . 1) * )) + ((p3 . 2) * )) + ((p3 . 3) * ) = ((((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * )) + ((((p3 . 1) * ) + ((p3 . 2) * )) + ((p3 . 3) * )) proof A6: ((p2 . 1) * ) . 1 = (p2 . 1) * 1 by A2, RVSUM_1:44 .= p2 . 1 ; A7: ((p2 . 1) * ) . 2 = (p2 . 1) * 0 by A2, RVSUM_1:44 .= 0 ; A8: ((p2 . 1) * ) . 3 = (p2 . 1) * 0 by A2, RVSUM_1:44 .= 0 ; A9: ((p2 . 2) * ) . 1 = (p2 . 2) * ( . 1) by RVSUM_1:44 .= 0 by A3 ; A10: ((p2 . 2) * ) . 2 = (p2 . 2) * 1 by A3, RVSUM_1:44 .= p2 . 2 ; A11: ((p2 . 2) * ) . 3 = (p2 . 2) * 0 by A3, RVSUM_1:44 .= 0 ; A12: ((p2 . 3) * ) . 1 = (p2 . 3) * 0 by A4, RVSUM_1:44 .= 0 ; A13: ((p2 . 3) * ) . 2 = (p2 . 3) * 0 by A4, RVSUM_1:44 .= 0 ; A14: ((p2 . 3) * ) . 3 = (p2 . 3) * 1 by A4, RVSUM_1:44 .= p2 . 3 ; A15: ((p3 . 1) * ) . 1 = (p3 . 1) * 1 by A2, RVSUM_1:44 .= p3 . 1 ; A16: ((p3 . 1) * ) . 2 = (p3 . 1) * 0 by A2, RVSUM_1:44 .= 0 ; A17: ((p3 . 1) * ) . 3 = (p3 . 1) * 0 by A2, RVSUM_1:44 .= 0 ; A18: ((p3 . 2) * ) . 1 = (p3 . 2) * ( . 1) by RVSUM_1:44 .= 0 by A3 ; A19: ((p3 . 2) * ) . 2 = (p3 . 2) * 1 by A3, RVSUM_1:44 .= p3 . 2 ; A20: ((p3 . 2) * ) . 3 = (p3 . 2) * 0 by A3, RVSUM_1:44 .= 0 ; A21: ((p3 . 3) * ) . 1 = (p3 . 3) * 0 by A4, RVSUM_1:44 .= 0 ; A22: ((p3 . 3) * ) . 2 = (p3 . 3) * 0 by A4, RVSUM_1:44 .= 0 ; A23: ((p3 . 3) * ) . 3 = (p3 . 3) * 1 by A4, RVSUM_1:44 .= p3 . 3 ; A24: (((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * ) = ((p2 . 1) * ) + (((p2 . 2) * ) + ((p2 . 3) * )) by RVSUM_1:15 .= ((p2 . 1) * ) + |[(0 + 0),((p2 . 2) + 0),(0 + (p2 . 3))]| by A9, A10, A11, A12, A13, A14, Lm2 .= |[(p2 . 1),0,0]| + |[0,(p2 . 2),(p2 . 3)]| by A6, A7, A8, Th1 .= |[((p2 . 1) + 0),(0 + (p2 . 2)),(0 + (p2 . 3))]| by Lm8 .= |[(p2 . 1),(p2 . 2),(p2 . 3)]| ; A25: (((p3 . 1) * ) + ((p3 . 2) * )) + ((p3 . 3) * ) = ((p3 . 1) * ) + (((p3 . 2) * ) + ((p3 . 3) * )) by RVSUM_1:15 .= ((p3 . 1) * ) + |[(0 + 0),((p3 . 2) + 0),(0 + (p3 . 3))]| by A18, A19, A20, A21, A22, A23, Lm2 .= |[(p3 . 1),0,0]| + |[0,(p3 . 2),(p3 . 3)]| by A15, A16, A17, Th1 .= |[((p3 . 1) + 0),(0 + (p3 . 2)),(0 + (p3 . 3))]| by Lm8 .= |[(p3 . 1),(p3 . 2),(p3 . 3)]| ; ((((((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * )) + ((p3 . 1) * )) + ((p3 . 2) * )) + ((p3 . 3) * ) = (((((p2 . 1) * ) + (((p2 . 2) * ) + ((p2 . 3) * ))) + ((p3 . 1) * )) + ((p3 . 2) * )) + ((p3 . 3) * ) by RVSUM_1:15 .= (((((p2 . 1) * ) + |[(0 + 0),((p2 . 2) + 0),(0 + (p2 . 3))]|) + ((p3 . 1) * )) + ((p3 . 2) * )) + ((p3 . 3) * ) by A9, A10, A11, A12, A13, A14, Lm2 .= (((|[(p2 . 1),0,0]| + |[0,(p2 . 2),(p2 . 3)]|) + ((p3 . 1) * )) + ((p3 . 2) * )) + ((p3 . 3) * ) by A6, A7, A8, Th1 .= ((|[((p2 . 1) + 0),(0 + (p2 . 2)),(0 + (p2 . 3))]| + ((p3 . 1) * )) + ((p3 . 2) * )) + ((p3 . 3) * ) by Lm8 .= ((|[(p2 . 1),(p2 . 2),(p2 . 3)]| + |[(p3 . 1),0,0]|) + ((p3 . 2) * )) + ((p3 . 3) * ) by A15, A16, A17, Th1 .= (|[((p2 . 1) + (p3 . 1)),((p2 . 2) + 0),((p2 . 3) + 0)]| + ((p3 . 2) * )) + ((p3 . 3) * ) by Lm8 .= (|[((p2 . 1) + (p3 . 1)),(p2 . 2),(p2 . 3)]| + |[0,(p3 . 2),0]|) + ((p3 . 3) * ) by A18, A19, A20, Th1 .= |[(((p2 . 1) + (p3 . 1)) + 0),((p2 . 2) + (p3 . 2)),((p2 . 3) + 0)]| + ((p3 . 3) * ) by Lm8 .= |[((p2 . 1) + (p3 . 1)),((p2 . 2) + (p3 . 2)),(p2 . 3)]| + |[0,0,(p3 . 3)]| by A21, A22, A23, Th1 .= |[(((p2 . 1) + (p3 . 1)) + 0),(((p2 . 2) + (p3 . 2)) + 0),((p2 . 3) + (p3 . 3))]| by Lm8 ; hence ((((((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * )) + ((p3 . 1) * )) + ((p3 . 2) * )) + ((p3 . 3) * ) = ((((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * )) + ((((p3 . 1) * ) + ((p3 . 2) * )) + ((p3 . 3) * )) by A24, A25, Lm8; ::_thesis: verum end; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = (((((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * )) + ((p3 . 1) * )) + ((p3 . 2) * )) + ((p3 . 3) * ) by A5, EUCLIDLP:24; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = ((((((p1 . 1) - (p3 . 1)) * ) + ((((p1 . 2) - (p3 . 2)) * ) + (((p1 . 3) - (p3 . 3)) * ))) + ((p3 . 1) * )) + ((p3 . 2) * )) + ((p3 . 3) * ) by RVSUM_1:15; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = ((((((p1 . 1) * ) - ((p3 . 1) * )) + ((((p1 . 2) - (p3 . 2)) * ) + (((p1 . 3) - (p3 . 3)) * ))) + ((p3 . 1) * )) + ((p3 . 2) * )) + ((p3 . 3) * ) by A1; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = ((((((p1 . 1) * ) + ((((p1 . 2) - (p3 . 2)) * ) + (((p1 . 3) - (p3 . 3)) * ))) - ((p3 . 1) * )) + ((p3 . 1) * )) + ((p3 . 2) * )) + ((p3 . 3) * ) by RVSUM_1:15; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = (((((p1 . 1) * ) + ((((p1 . 2) - (p3 . 2)) * ) + (((p1 . 3) - (p3 . 3)) * ))) + (((p3 . 1) * ) - ((p3 . 1) * ))) + ((p3 . 2) * )) + ((p3 . 3) * ) by RVSUM_1:15; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = (((((p1 . 1) * ) + ((((p1 . 2) - (p3 . 2)) * ) + (((p1 . 3) - (p3 . 3)) * ))) + (0.REAL 3)) + ((p3 . 2) * )) + ((p3 . 3) * ) by RVSUM_1:37; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = ((((p1 . 1) * ) + ((((p1 . 2) - (p3 . 2)) * ) + (((p1 . 3) - (p3 . 3)) * ))) + ((p3 . 2) * )) + ((p3 . 3) * ) by EUCLID_4:1; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = (((((p1 . 1) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * )) + ((p3 . 2) * )) + ((p3 . 3) * ) by RVSUM_1:15; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = (((((p1 . 1) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) * ) - ((p3 . 3) * ))) + ((p3 . 2) * )) + ((p3 . 3) * ) by A1; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = ((((((p1 . 1) * ) + (((p1 . 2) - (p3 . 2)) * )) + ((p1 . 3) * )) - ((p3 . 3) * )) + ((p3 . 2) * )) + ((p3 . 3) * ) by RVSUM_1:15; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = ((((((p1 . 1) * ) + (((p1 . 2) - (p3 . 2)) * )) + ((p1 . 3) * )) + ((p3 . 2) * )) - ((p3 . 3) * )) + ((p3 . 3) * ) by RVSUM_1:15; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = (((((p1 . 1) * ) + (((p1 . 2) - (p3 . 2)) * )) + ((p1 . 3) * )) + ((p3 . 2) * )) + (((p3 . 3) * ) - ((p3 . 3) * )) by RVSUM_1:15; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = (((((p1 . 1) * ) + (((p1 . 2) - (p3 . 2)) * )) + ((p1 . 3) * )) + ((p3 . 2) * )) + (0.REAL 3) by RVSUM_1:37; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = ((((p1 . 1) * ) + (((p1 . 2) - (p3 . 2)) * )) + ((p1 . 3) * )) + ((p3 . 2) * ) by EUCLID_4:1; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = (((p1 . 1) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) * ) + ((p3 . 2) * )) by RVSUM_1:15; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = (((p1 . 1) * ) + (((p1 . 2) * ) - ((p3 . 2) * ))) + (((p1 . 3) * ) + ((p3 . 2) * )) by A1; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = ((((p1 . 1) * ) + ((p1 . 2) * )) + (- ((p3 . 2) * ))) + (((p1 . 3) * ) + ((p3 . 2) * )) by RVSUM_1:15; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = (((((p1 . 1) * ) + ((p1 . 2) * )) - ((p3 . 2) * )) + ((p1 . 3) * )) + ((p3 . 2) * ) by RVSUM_1:15; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = (((((p1 . 1) * ) + ((p1 . 2) * )) + ((p1 . 3) * )) - ((p3 . 2) * )) + ((p3 . 2) * ) by RVSUM_1:15; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = ((((p1 . 1) * ) + ((p1 . 2) * )) + ((p1 . 3) * )) + (((p3 . 2) * ) - ((p3 . 2) * )) by RVSUM_1:15; then ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) = ((((p1 . 1) * ) + ((p1 . 2) * )) + ((p1 . 3) * )) + (0.REAL 3) by RVSUM_1:37; hence ( (((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * ) = ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) implies (((p1 . 1) * ) + ((p1 . 2) * )) + ((p1 . 3) * ) = ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) ) by EUCLID_4:1; ::_thesis: verum end; hence ( (((p2 . 1) * ) + ((p2 . 2) * )) + ((p2 . 3) * ) = ((((p1 . 1) - (p3 . 1)) * ) + (((p1 . 2) - (p3 . 2)) * )) + (((p1 . 3) - (p3 . 3)) * ) implies (((p1 . 1) * ) + ((p1 . 2) * )) + ((p1 . 3) * ) = ((((p2 . 1) + (p3 . 1)) * ) + (((p2 . 2) + (p3 . 2)) * )) + (((p2 . 3) + (p3 . 3)) * ) ) ; ::_thesis: verum end; definition let f1, f2, f3 be PartFunc of REAL,REAL; func VFunc (f1,f2,f3) -> Function of REAL,(REAL 3) means :Def5: :: EUCLID_8:def 5 for t being Real holds it . t = |[(f1 . t),(f2 . t),(f3 . t)]|; existence ex b1 being Function of REAL,(REAL 3) st for t being Real holds b1 . t = |[(f1 . t),(f2 . t),(f3 . t)]| proof defpred S1[ set , set ] means \$2 = |[(f1 . \$1),(f2 . \$1),(f3 . \$1)]|; A1: for x being Element of REAL ex y being Element of REAL 3 st S1[x,y] ; consider F being Function of REAL,(REAL 3) such that A2: for t being Element of REAL holds S1[t,F . t] from FUNCT_2:sch_3(A1); take F ; ::_thesis: for t being Real holds F . t = |[(f1 . t),(f2 . t),(f3 . t)]| thus for t being Real holds F . t = |[(f1 . t),(f2 . t),(f3 . t)]| by A2; ::_thesis: verum end; uniqueness for b1, b2 being Function of REAL,(REAL 3) st ( for t being Real holds b1 . t = |[(f1 . t),(f2 . t),(f3 . t)]| ) & ( for t being Real holds b2 . t = |[(f1 . t),(f2 . t),(f3 . t)]| ) holds b1 = b2 proof let F, G be Function of REAL,(REAL 3); ::_thesis: ( ( for t being Real holds F . t = |[(f1 . t),(f2 . t),(f3 . t)]| ) & ( for t being Real holds G . t = |[(f1 . t),(f2 . t),(f3 . t)]| ) implies F = G ) assume that A3: for t being Real holds F . t = |[(f1 . t),(f2 . t),(f3 . t)]| and A4: for t being Real holds G . t = |[(f1 . t),(f2 . t),(f3 . t)]| ; ::_thesis: F = G now__::_thesis:_for_t_being_Element_of_REAL_holds_F_._t_=_G_._t let t be Element of REAL ; ::_thesis: F . t = G . t F . t = |[(f1 . t),(f2 . t),(f3 . t)]| by A3; hence F . t = G . t by A4; ::_thesis: verum end; hence F = G by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def5 defines VFunc EUCLID_8:def_5_:_ for f1, f2, f3 being PartFunc of REAL,REAL for b4 being Function of REAL,(REAL 3) holds ( b4 = VFunc (f1,f2,f3) iff for t being Real holds b4 . t = |[(f1 . t),(f2 . t),(f3 . t)]| ); theorem :: EUCLID_8:48 for f1, f2, f3 being PartFunc of REAL,REAL for t being Real holds (VFunc (f1,f2,f3)) . t = (((f1 . t) * ) + ((f2 . t) * )) + ((f3 . t) * ) proof let f1, f2, f3 be PartFunc of REAL,REAL; ::_thesis: for t being Real holds (VFunc (f1,f2,f3)) . t = (((f1 . t) * ) + ((f2 . t) * )) + ((f3 . t) * ) let t be Real; ::_thesis: (VFunc (f1,f2,f3)) . t = (((f1 . t) * ) + ((f2 . t) * )) + ((f3 . t) * ) A1: ( . 1 = 1 & . 2 = 0 & . 3 = 0 ) by FINSEQ_1:45; A2: ( . 1 = 0 & . 2 = 1 & . 3 = 0 ) by FINSEQ_1:45; A3: ( . 1 = 0 & . 2 = 0 & . 3 = 1 ) by FINSEQ_1:45; A4: (((f1 . t) * ) + ((f2 . t) * )) + ((f3 . t) * ) = (|[((f1 . t) * 1),((f1 . t) * 0),((f1 . t) * 0)]| + ((f2 . t) * )) + ((f3 . t) * ) by A1, Lm1 .= (|[(f1 . t),0,0]| + |[((f2 . t) * 0),((f2 . t) * 1),((f2 . t) * 0)]|) + ((f3 . t) * ) by A2, Lm1 .= (|[(f1 . t),0,0]| + |[0,(f2 . t),0]|) + |[((f3 . t) * 0),((f3 . t) * 0),((f3 . t) * 1)]| by A3, Lm1 .= (|[(f1 . t),0,0]| + |[0,(f2 . t),0]|) + |[0,0,(f3 . t)]| ; A5: ( |[(f1 . t),0,0]| . 1 = f1 . t & |[(f1 . t),0,0]| . 2 = 0 & |[(f1 . t),0,0]| . 3 = 0 ) by FINSEQ_1:45; A6: ( |[0,(f2 . t),0]| . 1 = 0 & |[0,(f2 . t),0]| . 2 = f2 . t & |[0,(f2 . t),0]| . 3 = 0 ) by FINSEQ_1:45; A7: ( |[0,0,(f3 . t)]| . 1 = 0 & |[0,0,(f3 . t)]| . 2 = 0 & |[0,0,(f3 . t)]| . 3 = f3 . t ) by FINSEQ_1:45; A8: (((f1 . t) * ) + ((f2 . t) * )) + ((f3 . t) * ) = |[((f1 . t) + 0),(0 + (f2 . t)),(0 + 0)]| + |[0,0,(f3 . t)]| by A4, A5, A6, Lm2 .= |[(f1 . t),(f2 . t),0]| + |[0,0,(f3 . t)]| ; ( |[(f1 . t),(f2 . t),0]| . 1 = f1 . t & |[(f1 . t),(f2 . t),0]| . 2 = f2 . t & |[(f1 . t),(f2 . t),0]| . 3 = 0 ) by FINSEQ_1:45; then (((f1 . t) * ) + ((f2 . t) * )) + ((f3 . t) * ) = |[((f1 . t) + 0),((f2 . t) + 0),(0 + (f3 . t))]| by A7, A8, Lm2 .= |[(f1 . t),(f2 . t),(f3 . t)]| ; hence (VFunc (f1,f2,f3)) . t = (((f1 . t) * ) + ((f2 . t) * )) + ((f3 . t) * ) by Def5; ::_thesis: verum end; theorem Th49: :: EUCLID_8:49 for p being Element of REAL 3 for f1, f2, f3 being PartFunc of REAL,REAL for t being Real holds ( p = (VFunc (f1,f2,f3)) . t iff ( p . 1 = f1 . t & p . 2 = f2 . t & p . 3 = f3 . t ) ) proof let p be Element of REAL 3; ::_thesis: for f1, f2, f3 being PartFunc of REAL,REAL for t being Real holds ( p = (VFunc (f1,f2,f3)) . t iff ( p . 1 = f1 . t & p . 2 = f2 . t & p . 3 = f3 . t ) ) let f1, f2, f3 be PartFunc of REAL,REAL; ::_thesis: for t being Real holds ( p = (VFunc (f1,f2,f3)) . t iff ( p . 1 = f1 . t & p . 2 = f2 . t & p . 3 = f3 . t ) ) let t be Real; ::_thesis: ( p = (VFunc (f1,f2,f3)) . t iff ( p . 1 = f1 . t & p . 2 = f2 . t & p . 3 = f3 . t ) ) thus ( p = (VFunc (f1,f2,f3)) . t implies ( p . 1 = f1 . t & p . 2 = f2 . t & p . 3 = f3 . t ) ) ::_thesis: ( p . 1 = f1 . t & p . 2 = f2 . t & p . 3 = f3 . t implies p = (VFunc (f1,f2,f3)) . t ) proof assume p = (VFunc (f1,f2,f3)) . t ; ::_thesis: ( p . 1 = f1 . t & p . 2 = f2 . t & p . 3 = f3 . t ) then p = |[(f1 . t),(f2 . t),(f3 . t)]| by Def5; hence ( p . 1 = f1 . t & p . 2 = f2 . t & p . 3 = f3 . t ) by FINSEQ_1:45; ::_thesis: verum end; assume ( p . 1 = f1 . t & p . 2 = f2 . t & p . 3 = f3 . t ) ; ::_thesis: p = (VFunc (f1,f2,f3)) . t then p = |[(f1 . t),(f2 . t),(f3 . t)]| by Th1; hence p = (VFunc (f1,f2,f3)) . t by Def5; ::_thesis: verum end; theorem Th50: :: EUCLID_8:50 for p being Element of REAL 3 holds ( len p = 3 & dom p = Seg 3 ) proof let p be Element of REAL 3; ::_thesis: ( len p = 3 & dom p = Seg 3 ) p = |[(p . 1),(p . 2),(p . 3)]| by Th1; hence ( len p = 3 & dom p = Seg 3 ) by FINSEQ_1:45, FINSEQ_1:89; ::_thesis: verum end; theorem Th51: :: EUCLID_8:51 for p, q being Element of REAL 3 holds mlt (p,q) = <*((p . 1) * (q . 1)),((p . 2) * (q . 2)),((p . 3) * (q . 3))*> proof let p, q be Element of REAL 3; ::_thesis: mlt (p,q) = <*((p . 1) * (q . 1)),((p . 2) * (q . 2)),((p . 3) * (q . 3))*> ( len p = 3 & len q = 3 ) by Th50; hence mlt (p,q) = <*((p . 1) * (q . 1)),((p . 2) * (q . 2)),((p . 3) * (q . 3))*> by EUCLID_5:28; ::_thesis: verum end; theorem :: EUCLID_8:52 for r being Element of REAL for p being Element of REAL 3 holds r * p = |[(r * (p . 1)),(r * (p . 2)),(r * (p . 3))]| by Lm1; theorem :: EUCLID_8:53 for p being Element of REAL 3 holds - p = |[(- (p . 1)),(- (p . 2)),(- (p . 3))]| proof let p be Element of REAL 3; ::_thesis: - p = |[(- (p . 1)),(- (p . 2)),(- (p . 3))]| - p = |[((- 1) * (p . 1)),((- 1) * (p . 2)),((- 1) * (p . 3))]| by Lm1 .= |[(- (p . 1)),(- (p . 2)),(- (p . 3))]| ; hence - p = |[(- (p . 1)),(- (p . 2)),(- (p . 3))]| ; ::_thesis: verum end; theorem :: EUCLID_8:54 for p being Element of REAL 3 holds len (- p) = len p proof let p be Element of REAL 3; ::_thesis: len (- p) = len p A1: len p = 3 by Th50; - p = |[((- p) . 1),((- p) . 2),((- p) . 3)]| by Th1; hence len (- p) = len p by A1, FINSEQ_1:45; ::_thesis: verum end; theorem :: EUCLID_8:55 for p, q being Element of REAL 3 holds p + q = |[((p . 1) + (q . 1)),((p . 2) + (q . 2)),((p . 3) + (q . 3))]| by Lm2; theorem :: EUCLID_8:56 for p, q being Element of REAL 3 for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t1, t2 being Real st p = (VFunc (f1,f2,f3)) . t1 & q = (VFunc (g1,g2,g3)) . t2 & p = q holds ( f1 . t1 = g1 . t2 & f2 . t1 = g2 . t2 & f3 . t1 = g3 . t2 ) proof let p, q be Element of REAL 3; ::_thesis: for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t1, t2 being Real st p = (VFunc (f1,f2,f3)) . t1 & q = (VFunc (g1,g2,g3)) . t2 & p = q holds ( f1 . t1 = g1 . t2 & f2 . t1 = g2 . t2 & f3 . t1 = g3 . t2 ) let f1, f2, f3, g1, g2, g3 be PartFunc of REAL,REAL; ::_thesis: for t1, t2 being Real st p = (VFunc (f1,f2,f3)) . t1 & q = (VFunc (g1,g2,g3)) . t2 & p = q holds ( f1 . t1 = g1 . t2 & f2 . t1 = g2 . t2 & f3 . t1 = g3 . t2 ) let t1, t2 be Real; ::_thesis: ( p = (VFunc (f1,f2,f3)) . t1 & q = (VFunc (g1,g2,g3)) . t2 & p = q implies ( f1 . t1 = g1 . t2 & f2 . t1 = g2 . t2 & f3 . t1 = g3 . t2 ) ) assume A1: ( p = (VFunc (f1,f2,f3)) . t1 & q = (VFunc (g1,g2,g3)) . t2 & p = q ) ; ::_thesis: ( f1 . t1 = g1 . t2 & f2 . t1 = g2 . t2 & f3 . t1 = g3 . t2 ) then A2: ( p . 1 = f1 . t1 & q . 1 = g1 . t2 ) by Th49; A3: ( p . 2 = f2 . t1 & q . 2 = g2 . t2 ) by A1, Th49; ( p . 3 = f3 . t1 & q . 3 = g3 . t2 ) by A1, Th49; hence ( f1 . t1 = g1 . t2 & f2 . t1 = g2 . t2 & f3 . t1 = g3 . t2 ) by A1, A2, A3; ::_thesis: verum end; theorem :: EUCLID_8:57 for f1, g1, f2, g2, f3, g3 being PartFunc of REAL,REAL for t1, t2 being Real st f1 . t1 = g1 . t2 & f2 . t1 = g2 . t2 & f3 . t1 = g3 . t2 holds (VFunc (f1,f2,f3)) . t1 = (VFunc (g1,g2,g3)) . t2 proof let f1, g1, f2, g2, f3, g3 be PartFunc of REAL,REAL; ::_thesis: for t1, t2 being Real st f1 . t1 = g1 . t2 & f2 . t1 = g2 . t2 & f3 . t1 = g3 . t2 holds (VFunc (f1,f2,f3)) . t1 = (VFunc (g1,g2,g3)) . t2 let t1, t2 be Real; ::_thesis: ( f1 . t1 = g1 . t2 & f2 . t1 = g2 . t2 & f3 . t1 = g3 . t2 implies (VFunc (f1,f2,f3)) . t1 = (VFunc (g1,g2,g3)) . t2 ) assume A1: ( f1 . t1 = g1 . t2 & f2 . t1 = g2 . t2 & f3 . t1 = g3 . t2 ) ; ::_thesis: (VFunc (f1,f2,f3)) . t1 = (VFunc (g1,g2,g3)) . t2 set p = |[(f1 . t1),(f2 . t1),(f3 . t1)]|; set q = |[(g1 . t2),(g2 . t2),(g3 . t2)]|; ( |[(f1 . t1),(f2 . t1),(f3 . t1)]| = (VFunc (f1,f2,f3)) . t1 & |[(g1 . t2),(g2 . t2),(g3 . t2)]| = (VFunc (g1,g2,g3)) . t2 ) by Def5; hence (VFunc (f1,f2,f3)) . t1 = (VFunc (g1,g2,g3)) . t2 by A1; ::_thesis: verum end; theorem Th58: :: EUCLID_8:58 for p being Element of REAL 3 for r being real number holds r * p = |[(r * (p . 1)),(r * (p . 2)),(r * (p . 3))]| proof let p be Element of REAL 3; ::_thesis: for r being real number holds r * p = |[(r * (p . 1)),(r * (p . 2)),(r * (p . 3))]| let r be real number ; ::_thesis: r * p = |[(r * (p . 1)),(r * (p . 2)),(r * (p . 3))]| A1: (r * p) . 1 = r * (p . 1) by RVSUM_1:44; A2: (r * p) . 2 = r * (p . 2) by RVSUM_1:44; (r * p) . 3 = r * (p . 3) by RVSUM_1:44; hence r * p = |[(r * (p . 1)),(r * (p . 2)),(r * (p . 3))]| by A1, A2, Th1; ::_thesis: verum end; theorem Th59: :: EUCLID_8:59 for x, y, z being Element of REAL for r being real number holds r * |[x,y,z]| = |[(r * x),(r * y),(r * z)]| proof let x, y, z be Element of REAL ; ::_thesis: for r being real number holds r * |[x,y,z]| = |[(r * x),(r * y),(r * z)]| let r be real number ; ::_thesis: r * |[x,y,z]| = |[(r * x),(r * y),(r * z)]| set p = |[x,y,z]|; r * |[x,y,z]| = |[(r * (|[x,y,z]| . 1)),(r * (|[x,y,z]| . 2)),(r * (|[x,y,z]| . 3))]| by Th58 .= |[(r * x),(r * (|[x,y,z]| . 2)),(r * (|[x,y,z]| . 3))]| by FINSEQ_1:45 .= |[(r * x),(r * y),(r * (|[x,y,z]| . 3))]| by FINSEQ_1:45 .= |[(r * x),(r * y),(r * z)]| by FINSEQ_1:45 ; hence r * |[x,y,z]| = |[(r * x),(r * y),(r * z)]| ; ::_thesis: verum end; theorem Th60: :: EUCLID_8:60 for p being Element of REAL 3 holds - p = |[(- (p . 1)),(- (p . 2)),(- (p . 3))]| proof let p be Element of REAL 3; ::_thesis: - p = |[(- (p . 1)),(- (p . 2)),(- (p . 3))]| reconsider r = - 1 as Element of REAL by XREAL_0:def_1; r * p = |[((- 1) * (p . 1)),((- 1) * (p . 2)),((- 1) * (p . 3))]| by Th58 .= |[(- (p . 1)),(- (p . 2)),(- (p . 3))]| ; hence - p = |[(- (p . 1)),(- (p . 2)),(- (p . 3))]| ; ::_thesis: verum end; theorem Th61: :: EUCLID_8:61 for p being Element of REAL 3 holds ( (- p) . 1 = - (p . 1) & (- p) . 2 = - (p . 2) & (- p) . 3 = - (p . 3) ) proof let p be Element of REAL 3; ::_thesis: ( (- p) . 1 = - (p . 1) & (- p) . 2 = - (p . 2) & (- p) . 3 = - (p . 3) ) - p = |[(- (p . 1)),(- (p . 2)),(- (p . 3))]| by Th60; hence ( (- p) . 1 = - (p . 1) & (- p) . 2 = - (p . 2) & (- p) . 3 = - (p . 3) ) by FINSEQ_1:45; ::_thesis: verum end; theorem :: EUCLID_8:62 for p1, p2 being Element of REAL 3 holds p1 - p2 = |[((p1 . 1) - (p2 . 1)),((p1 . 2) - (p2 . 2)),((p1 . 3) - (p2 . 3))]| proof let p1, p2 be Element of REAL 3; ::_thesis: p1 - p2 = |[((p1 . 1) - (p2 . 1)),((p1 . 2) - (p2 . 2)),((p1 . 3) - (p2 . 3))]| A1: (- p2) . 1 = - (p2 . 1) by Th61; A2: (- p2) . 2 = - (p2 . 2) by Th61; (- p2) . 3 = - (p2 . 3) by Th61; then p1 + (- p2) = |[((p1 . 1) + (- (p2 . 1))),((p1 . 2) + (- (p2 . 2))),((p1 . 3) + (- (p2 . 3)))]| by A1, A2, Lm2; hence p1 - p2 = |[((p1 . 1) - (p2 . 1)),((p1 . 2) - (p2 . 2)),((p1 . 3) - (p2 . 3))]| ; ::_thesis: verum end; theorem :: EUCLID_8:63 for p, q being Element of REAL 3 holds |(p,q)| = (((p . 1) * (q . 1)) + ((p . 2) * (q . 2))) + ((p . 3) * (q . 3)) by Lm5; theorem Th64: :: EUCLID_8:64 for p being Element of REAL 3 holds |(p,p)| = (((p . 1) ^2) + ((p . 2) ^2)) + ((p . 3) ^2) proof let p be Element of REAL 3; ::_thesis: |(p,p)| = (((p . 1) ^2) + ((p . 2) ^2)) + ((p . 3) ^2) p = |[(p . 1),(p . 2),(p . 3)]| by Th1; hence |(p,p)| = (((p . 1) ^2) + ((p . 2) ^2)) + ((p . 3) ^2) by Lm12; ::_thesis: verum end; theorem Th65: :: EUCLID_8:65 for p being Element of REAL 3 holds |.p.| = sqrt ((((p . 1) ^2) + ((p . 2) ^2)) + ((p . 3) ^2)) proof let p be Element of REAL 3; ::_thesis: |.p.| = sqrt ((((p . 1) ^2) + ((p . 2) ^2)) + ((p . 3) ^2)) |.p.| = sqrt (Sum <*((p . 1) * (p . 1)),((p . 2) * (p . 2)),((p . 3) * (p . 3))*>) by Th51 .= sqrt ((((p . 1) ^2) + ((p . 2) ^2)) + ((p . 3) ^2)) by RVSUM_1:78 ; hence |.p.| = sqrt ((((p . 1) ^2) + ((p . 2) ^2)) + ((p . 3) ^2)) ; ::_thesis: verum end; theorem :: EUCLID_8:66 for r being Element of REAL for p being Element of REAL 3 holds |.(r * p).| = (abs r) * (sqrt ((((p . 1) ^2) + ((p . 2) ^2)) + ((p . 3) ^2))) proof let r be Element of REAL ; ::_thesis: for p being Element of REAL 3 holds |.(r * p).| = (abs r) * (sqrt ((((p . 1) ^2) + ((p . 2) ^2)) + ((p . 3) ^2))) let p be Element of REAL 3; ::_thesis: |.(r * p).| = (abs r) * (sqrt ((((p . 1) ^2) + ((p . 2) ^2)) + ((p . 3) ^2))) |.(r * p).| = (abs r) * |.p.| by EUCLID:11 .= (abs r) * (sqrt ((((p . 1) ^2) + ((p . 2) ^2)) + ((p . 3) ^2))) by Th65 ; hence |.(r * p).| = (abs r) * (sqrt ((((p . 1) ^2) + ((p . 2) ^2)) + ((p . 3) ^2))) ; ::_thesis: verum end; theorem :: EUCLID_8:67 for r1, r2 being Element of REAL for p being Element of REAL 3 holds (r1 * p) + (r2 * p) = (r1 + r2) * |[(p . 1),(p . 2),(p . 3)]| proof let r1, r2 be Element of REAL ; ::_thesis: for p being Element of REAL 3 holds (r1 * p) + (r2 * p) = (r1 + r2) * |[(p . 1),(p . 2),(p . 3)]| let p be Element of REAL 3; ::_thesis: (r1 * p) + (r2 * p) = (r1 + r2) * |[(p . 1),(p . 2),(p . 3)]| (r1 * p) + (r2 * p) = (r1 + r2) * p by RVSUM_1:50; hence (r1 * p) + (r2 * p) = (r1 + r2) * |[(p . 1),(p . 2),(p . 3)]| by Th1; ::_thesis: verum end; theorem :: EUCLID_8:68 for r being Element of REAL for p1, p2 being Element of REAL 3 holds |((r * p1),p2)| = r * |(p1,p2)| by RVSUM_1:131; theorem :: EUCLID_8:69 for r1, r2 being Element of REAL for p being Element of REAL 3 holds (r1 * p) - (r2 * p) = (r1 - r2) * |[(p . 1),(p . 2),(p . 3)]| proof let r1, r2 be Element of REAL ; ::_thesis: for p being Element of REAL 3 holds (r1 * p) - (r2 * p) = (r1 - r2) * |[(p . 1),(p . 2),(p . 3)]| let p be Element of REAL 3; ::_thesis: (r1 * p) - (r2 * p) = (r1 - r2) * |[(p . 1),(p . 2),(p . 3)]| for R being Element of 3 -tuples_on REAL holds (r1 * R) - (r2 * R) = (r1 - r2) * R proof let R be Element of 3 -tuples_on REAL; ::_thesis: (r1 * R) - (r2 * R) = (r1 - r2) * R (r1 * R) - (r2 * R) = (r1 * R) + (((- 1) * r2) * R) by RVSUM_1:49 .= (r1 + (- r2)) * R by RVSUM_1:50 ; hence (r1 * R) - (r2 * R) = (r1 - r2) * R ; ::_thesis: verum end; then (r1 * p) - (r2 * p) = (r1 - r2) * p ; hence (r1 * p) - (r2 * p) = (r1 - r2) * |[(p . 1),(p . 2),(p . 3)]| by Th1; ::_thesis: verum end; theorem :: EUCLID_8:70 for r being Element of REAL for p, q being Element of REAL 3 holds |((r * p),q)| = r * ((((p . 1) * (q . 1)) + ((p . 2) * (q . 2))) + ((p . 3) * (q . 3))) proof let r be Element of REAL ; ::_thesis: for p, q being Element of REAL 3 holds |((r * p),q)| = r * ((((p . 1) * (q . 1)) + ((p . 2) * (q . 2))) + ((p . 3) * (q . 3))) let p, q be Element of REAL 3; ::_thesis: |((r * p),q)| = r * ((((p . 1) * (q . 1)) + ((p . 2) * (q . 2))) + ((p . 3) * (q . 3))) |((r * p),q)| = r * |(p,q)| by RVSUM_1:131; hence |((r * p),q)| = r * ((((p . 1) * (q . 1)) + ((p . 2) * (q . 2))) + ((p . 3) * (q . 3))) by Lm5; ::_thesis: verum end; theorem :: EUCLID_8:71 for p being Element of REAL 3 holds |(p,(0.REAL 3))| = 0 proof let p be Element of REAL 3; ::_thesis: |(p,(0.REAL 3))| = 0 0.REAL 3 = |[0,0,0]| by FINSEQ_2:62; hence |(p,(0.REAL 3))| = 0 by Th13; ::_thesis: verum end; theorem :: EUCLID_8:72 for p1, p2 being Element of REAL 3 holds |((- p1),p2)| = - |(p1,p2)| by RVSUM_1:132; theorem :: EUCLID_8:73 for p1, p2 being Element of REAL 3 holds |((- p1),(- p2))| = |(p1,p2)| by RVSUM_1:133; theorem :: EUCLID_8:74 for p1, p2, q being Element of REAL 3 holds |((p1 - p2),q)| = |(p1,q)| - |(p2,q)| by RVSUM_1:134; theorem :: EUCLID_8:75 for p1, p2, q being Element of REAL 3 holds |((p1 + p2),q)| = |(p1,q)| + |(p2,q)| by RVSUM_1:130; theorem :: EUCLID_8:76 for r1, r2 being Element of REAL for p1, p2, q being Element of REAL 3 holds |(((r1 * p1) + (r2 * p2)),q)| = (r1 * |(p1,q)|) + (r2 * |(p2,q)|) by RVSUM_1:135; theorem :: EUCLID_8:77 for p1, p2, q1, q2 being Element of REAL 3 holds |((p1 + p2),(q1 + q2))| = ((|(p1,q1)| + |(p1,q2)|) + |(p2,q1)|) + |(p2,q2)| by RVSUM_1:136; theorem :: EUCLID_8:78 for p1, p2, q1, q2 being Element of REAL 3 holds |((p1 - p2),(q1 - q2))| = ((|(p1,q1)| - |(p1,q2)|) - |(p2,q1)|) + |(p2,q2)| by RVSUM_1:137; theorem Th79: :: EUCLID_8:79 for p being Element of REAL 3 holds ( |(p,p)| = 0 iff p = 0.REAL 3 ) proof let p be Element of REAL 3; ::_thesis: ( |(p,p)| = 0 iff p = 0.REAL 3 ) thus ( |(p,p)| = 0 implies p = 0.REAL 3 ) ::_thesis: ( p = 0.REAL 3 implies |(p,p)| = 0 ) proof assume |(p,p)| = 0 ; ::_thesis: p = 0.REAL 3 then Sum (sqr p) = 0 ; hence p = 0.REAL 3 by RVSUM_1:91; ::_thesis: verum end; assume p = 0.REAL 3 ; ::_thesis: |(p,p)| = 0 then A1: p = |[0,0,0]| by FINSEQ_2:62; then A2: p . 1 = 0 by FINSEQ_1:45; A3: p . 2 = 0 by A1, FINSEQ_1:45; p . 3 = 0 by A1, FINSEQ_1:45; then |(p,p)| = ((0 ^2) + (0 ^2)) + (0 ^2) by A2, A3, Th64; hence |(p,p)| = 0 ; ::_thesis: verum end; theorem :: EUCLID_8:80 for p being Element of REAL 3 holds ( |.p.| = 0 iff p = 0.REAL 3 ) proof let p be Element of REAL 3; ::_thesis: ( |.p.| = 0 iff p = 0.REAL 3 ) thus ( |.p.| = 0 implies p = 0.REAL 3 ) ::_thesis: ( p = 0.REAL 3 implies |.p.| = 0 ) proof assume A1: |.p.| = 0 ; ::_thesis: p = 0.REAL 3 |(p,p)| >= 0 by RVSUM_1:119; then (sqrt |(p,p)|) ^2 = |(p,p)| by SQUARE_1:def_2; hence p = 0.REAL 3 by A1, Th79; ::_thesis: verum end; assume p = 0.REAL 3 ; ::_thesis: |.p.| = 0 then A2: p = |[0,0,0]| by FINSEQ_2:62; then A3: p . 1 = 0 by FINSEQ_1:45; A4: p . 2 = 0 by A2, FINSEQ_1:45; p . 3 = 0 by A2, FINSEQ_1:45; then |(p,p)| = ((0 ^2) + (0 ^2)) + (0 ^2) by A3, A4, Th64; hence |.p.| = 0 by SQUARE_1:17; ::_thesis: verum end; theorem :: EUCLID_8:81 for p, q being Element of REAL 3 holds |((p - q),(p - q))| = (|(p,p)| - (2 * |(p,q)|)) + |(q,q)| by RVSUM_1:139; theorem :: EUCLID_8:82 for p, q being Element of REAL 3 holds |((p + q),(p + q))| = (|(p,p)| + (2 * |(p,q)|)) + |(q,q)| by RVSUM_1:138; theorem Th83: :: EUCLID_8:83 for p1, p2 being Element of REAL 3 holds p1 p2 = - (p2 p1) proof let p1, p2 be Element of REAL 3; ::_thesis: p1 p2 = - (p2 p1) - (p2 p1) = |[((- 1) * (((p2 . 2) * (p1 . 3)) - ((p2 . 3) * (p1 . 2)))),((- 1) * (((p2 . 3) * (p1 . 1)) - ((p2 . 1) * (p1 . 3)))),((- 1) * (((p2 . 1) * (p1 . 2)) - ((p2 . 2) * (p1 . 1))))]| by Th59 .= p1 p2 ; hence p1 p2 = - (p2 p1) ; ::_thesis: verum end; theorem Th84: :: EUCLID_8:84 for x1, x2, x3, y1, y2, y3 being Element of REAL holds |[x1,x2,x3]| |[y1,y2,y3]| = |[((x2 * y3) - (x3 * y2)),((x3 * y1) - (x1 * y3)),((x1 * y2) - (x2 * y1))]| proof let x1, x2, x3, y1, y2, y3 be Element of REAL ; ::_thesis: |[x1,x2,x3]| |[y1,y2,y3]| = |[((x2 * y3) - (x3 * y2)),((x3 * y1) - (x1 * y3)),((x1 * y2) - (x2 * y1))]| set p1 = |[x1,x2,x3]|; A1: ( |[x1,x2,x3]| . 1 = x1 & |[x1,x2,x3]| . 2 = x2 & |[x1,x2,x3]| . 3 = x3 ) by FINSEQ_1:45; set p2 = |[y1,y2,y3]|; ( |[y1,y2,y3]| . 1 = y1 & |[y1,y2,y3]| . 2 = y2 & |[y1,y2,y3]| . 3 = y3 ) by FINSEQ_1:45; hence |[x1,x2,x3]| |[y1,y2,y3]| = |[((x2 * y3) - (x3 * y2)),((x3 * y1) - (x1 * y3)),((x1 * y2) - (x2 * y1))]| by A1; ::_thesis: verum end; theorem :: EUCLID_8:85 for r being Element of REAL for p1, p2 being Element of REAL 3 holds ( (r * p1) p2 = r * (p1 p2) & (r * p1) p2 = p1 (r * p2) ) proof let r be Element of REAL ; ::_thesis: for p1, p2 being Element of REAL 3 holds ( (r * p1) p2 = r * (p1 p2) & (r * p1) p2 = p1 (r * p2) ) let p1, p2 be Element of REAL 3; ::_thesis: ( (r * p1) p2 = r * (p1 p2) & (r * p1) p2 = p1 (r * p2) ) A1: (r * p1) p2 = |[(r * (p1 . 1)),(r * (p1 . 2)),(r * (p1 . 3))]| p2 by Th58 .= |[(r * (p1 . 1)),(r * (p1 . 2)),(r * (p1 . 3))]| |[(p2 . 1),(p2 . 2),(p2 . 3)]| by Th1 .= |[(((r * (p1 . 2)) * (p2 . 3)) - ((r * (p1 . 3)) * (p2 . 2))),(((r * (p1 . 3)) * (p2 . 1)) - ((r * (p1 . 1)) * (p2 . 3))),(((r * (p1 . 1)) * (p2 . 2)) - ((r * (p1 . 2)) * (p2 . 1)))]| by Th84 ; then A2: (r * p1) p2 = |[(r * (((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2)))),(r * (((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3)))),(r * (((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1))))]| .= r * (p1 p2) by Th59 ; (r * p1) p2 = |[(((p1 . 2) * (r * (p2 . 3))) - ((p1 . 3) * (r * (p2 . 2)))),(((p1 . 3) * (r * (p2 . 1))) - ((p1 . 1) * (r * (p2 . 3)))),(((p1 . 1) * (r * (p2 . 2))) - ((p1 . 2) * (r * (p2 . 1))))]| by A1 .= |[(p1 . 1),(p1 . 2),(p1 . 3)]| |[(r * (p2 . 1)),(r * (p2 . 2)),(r * (p2 . 3))]| by Th84 .= p1 |[(r * (p2 . 1)),(r * (p2 . 2)),(r * (p2 . 3))]| by Th1 .= p1 (r * p2) by Th58 ; hence ( (r * p1) p2 = r * (p1 p2) & (r * p1) p2 = p1 (r * p2) ) by A2; ::_thesis: verum end; theorem Th86: :: EUCLID_8:86 for p1, p2, p3 being Element of REAL 3 holds p1 (p2 + p3) = (p1 p2) + (p1 p3) proof let p1, p2, p3 be Element of REAL 3; ::_thesis: p1 (p2 + p3) = (p1 p2) + (p1 p3) A1: p2 + p3 = |[((p2 . 1) + (p3 . 1)),((p2 . 2) + (p3 . 2)),((p2 . 3) + (p3 . 3))]| by Lm2; then A2: (p2 + p3) . 1 = (p2 . 1) + (p3 . 1) by FINSEQ_1:45; A3: (p2 + p3) . 2 = (p2 . 2) + (p3 . 2) by A1, FINSEQ_1:45; A4: (p2 + p3) . 3 = (p2 . 3) + (p3 . 3) by A1, FINSEQ_1:45; (p1 p2) + (p1 p3) = |[((((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2))) + (((p1 . 2) * (p3 . 3)) - ((p1 . 3) * (p3 . 2)))),((((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))) + (((p1 . 3) * (p3 . 1)) - ((p1 . 1) * (p3 . 3)))),((((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1))) + (((p1 . 1) * (p3 . 2)) - ((p1 . 2) * (p3 . 1))))]| by Lm8 .= |[(((((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2))) + ((p1 . 2) * (p3 . 3))) - ((p1 . 3) * (p3 . 2))),(((((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))) + ((p1 . 3) * (p3 . 1))) - ((p1 . 1) * (p3 . 3))),(((((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1))) + ((p1 . 1) * (p3 . 2))) - ((p1 . 2) * (p3 . 1)))]| ; hence p1 (p2 + p3) = (p1 p2) + (p1 p3) by A2, A3, A4; ::_thesis: verum end; theorem Th87: :: EUCLID_8:87 for p1, p2, p3 being Element of REAL 3 holds (p1 + p2) p3 = (p1 p3) + (p2 p3) proof let p1, p2, p3 be Element of REAL 3; ::_thesis: (p1 + p2) p3 = (p1 p3) + (p2 p3) (p1 + p2) p3 = - (p3 (p1 + p2)) by Th83 .= - ((p3 p1) + (p3 p2)) by Th86 .= - ((p3 p1) - (p2 p3)) by Th83 .= (- (p3 p1)) + (p2 p3) by RVSUM_1:36 ; hence (p1 + p2) p3 = (p1 p3) + (p2 p3) by Th83; ::_thesis: verum end; theorem :: EUCLID_8:88 for p1, p2, q1, q2 being Element of REAL 3 holds (p1 + p2) (q1 + q2) = (((p1 q1) + (p1 q2)) + (p2 q1)) + (p2 q2) proof let p1, p2, q1, q2 be Element of REAL 3; ::_thesis: (p1 + p2) (q1 + q2) = (((p1 q1) + (p1 q2)) + (p2 q1)) + (p2 q2) (p1 + p2) (q1 + q2) = (p1 (q1 + q2)) + (p2 (q1 + q2)) by Th87; then (p1 + p2) (q1 + q2) = ((p1 q1) + (p1 q2)) + (p2 (q1 + q2)) by Th86; then (p1 + p2) (q1 + q2) = ((p1 q1) + (p1 q2)) + ((p2 q1) + (p2 q2)) by Th86; hence (p1 + p2) (q1 + q2) = (((p1 q1) + (p1 q2)) + (p2 q1)) + (p2 q2) by RVSUM_1:15; ::_thesis: verum end; theorem :: EUCLID_8:89 for p1, p2, p3 being Element of REAL 3 holds p1 (p2 p3) = (|(p1,p3)| * p2) - (|(p1,p2)| * p3) proof let p1, p2, p3 be Element of REAL 3; ::_thesis: p1 (p2 p3) = (|(p1,p3)| * p2) - (|(p1,p2)| * p3) A1: p2 = |[(p2 . 1),(p2 . 2),(p2 . 3)]| by Th1; A2: p3 = |[(p3 . 1),(p3 . 2),(p3 . 3)]| by Th1; p1 (p2 p3) = |[(p1 . 1),(p1 . 2),(p1 . 3)]| |[(((p2 . 2) * (p3 . 3)) - ((p2 . 3) * (p3 . 2))),(((p2 . 3) * (p3 . 1)) - ((p2 . 1) * (p3 . 3))),(((p2 . 1) * (p3 . 2)) - ((p2 . 2) * (p3 . 1)))]| by Th1; then p1 (p2 p3) = |[(((p1 . 2) * (((p2 . 1) * (p3 . 2)) - ((p2 . 2) * (p3 . 1)))) - ((p1 . 3) * (((p2 . 3) * (p3 . 1)) - ((p2 . 1) * (p3 . 3))))),(((p1 . 3) * (((p2 . 2) * (p3 . 3)) - ((p2 . 3) * (p3 . 2)))) - ((p1 . 1) * (((p2 . 1) * (p3 . 2)) - ((p2 . 2) * (p3 . 1))))),(((p1 . 1) * (((p2 . 3) * (p3 . 1)) - ((p2 . 1) * (p3 . 3)))) - ((p1 . 2) * (((p2 . 2) * (p3 . 3)) - ((p2 . 3) * (p3 . 2)))))]| by Th84; then A3: p1 (p2 p3) = |[((((((p1 . 2) * (p3 . 2)) + ((p1 . 3) * (p3 . 3))) + ((p1 . 1) * (p3 . 1))) * (p2 . 1)) - (((((p1 . 2) * (p2 . 2)) + ((p1 . 3) * (p2 . 3))) + ((p1 . 1) * (p2 . 1))) * (p3 . 1))),((((((p1 . 3) * (p3 . 3)) + ((p1 . 1) * (p3 . 1))) + ((p1 . 2) * (p3 . 2))) * (p2 . 2)) - (((((p1 . 3) * (p2 . 3)) + ((p1 . 1) * (p2 . 1))) + ((p1 . 2) * (p2 . 2))) * (p3 . 2))),((((((p1 . 1) * (p3 . 1)) + ((p1 . 2) * (p3 . 2))) + ((p1 . 3) * (p3 . 3))) * (p2 . 3)) - (((((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (p2 . 3))) * (p3 . 3)))]| ; |(p1,p3)| = (((p1 . 1) * (p3 . 1)) + ((p1 . 2) * (p3 . 2))) + ((p1 . 3) * (p3 . 3)) by Lm5; then A4: p1 (p2 p3) = |[(|(p1,p3)| * (p2 . 1)),(|(p1,p3)| * (p2 . 2)),(|(p1,p3)| * (p2 . 3))]| - |[(((((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (p2 . 3))) * (p3 . 1)),(((((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (p2 . 3))) * (p3 . 2)),(((((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (p2 . 3))) * (p3 . 3))]| by A3, Lm11; |(p1,p2)| = (((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (p2 . 3)) by Lm5; then p1 (p2 p3) = (|(p1,p3)| * |[(p2 . 1),(p2 . 2),(p2 . 3)]|) - |[(|(p1,p2)| * (p3 . 1)),(|(p1,p2)| * (p3 . 2)),(|(p1,p2)| * (p3 . 3))]| by A4, Th59; hence p1 (p2 p3) = (|(p1,p3)| * p2) - (|(p1,p2)| * p3) by A1, A2, Th59; ::_thesis: verum end; definition let p1, p2, p3 be Element of REAL 3; func|{p1,p2,p3}| -> real number equals :: EUCLID_8:def 6 |(p1,(p2 p3))|; coherence |(p1,(p2 p3))| is real number ; end; :: deftheorem defines |{ EUCLID_8:def_6_:_ for p1, p2, p3 being Element of REAL 3 holds |{p1,p2,p3}| = |(p1,(p2 p3))|; theorem :: EUCLID_8:90 for p1, p2 being Element of REAL 3 holds |{p1,p1,p2}| = 0 proof let p1, p2 be Element of REAL 3; ::_thesis: |{p1,p1,p2}| = 0 A1: (p1 p2) . 1 = ((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2)) by FINSEQ_1:45; A2: (p1 p2) . 2 = ((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3)) by FINSEQ_1:45; (p1 p2) . 3 = ((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1)) by FINSEQ_1:45; then |{p1,p1,p2}| = (((p1 . 1) * (((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2)))) + ((p1 . 2) * (((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))))) + ((p1 . 3) * (((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1)))) by A2, A1, Lm5 .= 0 ; hence |{p1,p1,p2}| = 0 ; ::_thesis: verum end; theorem :: EUCLID_8:91 for p2, p1 being Element of REAL 3 holds |{p2,p1,p2}| = 0 proof let p2, p1 be Element of REAL 3; ::_thesis: |{p2,p1,p2}| = 0 A1: (p1 p2) . 1 = ((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2)) by FINSEQ_1:45; A2: (p1 p2) . 2 = ((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3)) by FINSEQ_1:45; (p1 p2) . 3 = ((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1)) by FINSEQ_1:45; then |{p2,p1,p2}| = (((p2 . 1) * (((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2)))) + ((p2 . 2) * (((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))))) + ((p2 . 3) * (((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1)))) by A2, A1, Lm5 .= 0 ; hence |{p2,p1,p2}| = 0 ; ::_thesis: verum end; theorem :: EUCLID_8:92 for p1, p2 being Element of REAL 3 holds |{p1,p2,p2}| = 0 proof let p1, p2 be Element of REAL 3; ::_thesis: |{p1,p2,p2}| = 0 |{p1,p2,p2}| = (((p1 . 1) * ((p2 p2) . 1)) + ((p1 . 2) * ((p2 p2) . 2))) + ((p1 . 3) * ((p2 p2) . 3)) by Lm5 .= (((p1 . 1) * (((p2 . 2) * (p2 . 3)) - ((p2 . 3) * (p2 . 2)))) + ((p1 . 2) * ((p2 p2) . 2))) + ((p1 . 3) * ((p2 p2) . 3)) by FINSEQ_1:45 .= (((p1 . 1) * (((p2 . 2) * (p2 . 3)) - ((p2 . 3) * (p2 . 2)))) + ((p1 . 2) * (((p2 . 3) * (p2 . 1)) - ((p2 . 1) * (p2 . 3))))) + ((p1 . 3) * ((p2 p2) . 3)) by FINSEQ_1:45 .= ((((p1 . 1) * ((p2 . 2) * (p2 . 3))) - ((p1 . 1) * ((p2 . 3) * (p2 . 2)))) + ((p1 . 2) * (((p2 . 3) * (p2 . 1)) - ((p2 . 1) * (p2 . 3))))) + ((p1 . 3) * (((p2 . 1) * (p2 . 2)) - ((p2 . 2) * (p2 . 1)))) by FINSEQ_1:45 .= (0 - ((p2 . 2) * ((p2 . 1) * (p2 . 3)))) + ((p2 . 2) * ((p2 . 1) * (p2 . 3))) ; hence |{p1,p2,p2}| = 0 ; ::_thesis: verum end; theorem Th93: :: EUCLID_8:93 for p1, p2, p3 being Element of REAL 3 holds |{p1,p2,p3}| = |{p2,p3,p1}| proof let p1, p2, p3 be Element of REAL 3; ::_thesis: |{p1,p2,p3}| = |{p2,p3,p1}| |{p1,p2,p3}| = |(|[(p1 . 1),(p1 . 2),(p1 . 3)]|,|[(((p2 . 2) * (p3 . 3)) - ((p2 . 3) * (p3 . 2))),(((p2 . 3) * (p3 . 1)) - ((p2 . 1) * (p3 . 3))),(((p2 . 1) * (p3 . 2)) - ((p2 . 2) * (p3 . 1)))]|)| by Th1 .= (((p1 . 1) * (((p2 . 2) * (p3 . 3)) - ((p2 . 3) * (p3 . 2)))) + ((p1 . 2) * (((p2 . 3) * (p3 . 1)) - ((p2 . 1) * (p3 . 3))))) + ((p1 . 3) * (((p2 . 1) * (p3 . 2)) - ((p2 . 2) * (p3 . 1)))) by EUCLID_5:30 .= (((p2 . 1) * (((p3 . 2) * (p1 . 3)) - ((p3 . 3) * (p1 . 2)))) + ((p2 . 2) * (((p3 . 3) * (p1 . 1)) - ((p3 . 1) * (p1 . 3))))) + ((p2 . 3) * (((p3 . 1) * (p1 . 2)) - ((p3 . 2) * (p1 . 1)))) .= |(|[(p2 . 1),(p2 . 2),(p2 . 3)]|,|[(((p3 . 2) * (p1 . 3)) - ((p3 . 3) * (p1 . 2))),(((p3 . 3) * (p1 . 1)) - ((p3 . 1) * (p1 . 3))),(((p3 . 1) * (p1 . 2)) - ((p3 . 2) * (p1 . 1)))]|)| by EUCLID_5:30 .= |(p2,(p3 p1))| by Th1 ; hence |{p1,p2,p3}| = |{p2,p3,p1}| ; ::_thesis: verum end; theorem :: EUCLID_8:94 for p1, p2, p3 being Element of REAL 3 holds |{p1,p2,p3}| = |((p1 p2),p3)| proof let p1, p2, p3 be Element of REAL 3; ::_thesis: |{p1,p2,p3}| = |((p1 p2),p3)| |{p1,p2,p3}| = |(|[(p1 . 1),(p1 . 2),(p1 . 3)]|,|[(((p2 . 2) * (p3 . 3)) - ((p2 . 3) * (p3 . 2))),(((p2 . 3) * (p3 . 1)) - ((p2 . 1) * (p3 . 3))),(((p2 . 1) * (p3 . 2)) - ((p2 . 2) * (p3 . 1)))]|)| by Th1 .= (((p1 . 1) * (((p2 . 2) * (p3 . 3)) - ((p2 . 3) * (p3 . 2)))) + ((p1 . 2) * (((p2 . 3) * (p3 . 1)) - ((p2 . 1) * (p3 . 3))))) + ((p1 . 3) * (((p2 . 1) * (p3 . 2)) - ((p2 . 2) * (p3 . 1)))) by EUCLID_5:30 .= ((((p2 . 2) * ((p1 . 1) * (p3 . 3))) - ((p2 . 3) * ((p1 . 1) * (p3 . 2)))) + (((p2 . 3) * ((p1 . 2) * (p3 . 1))) - ((p2 . 1) * ((p1 . 2) * (p3 . 3))))) + (((p2 . 1) * ((p1 . 3) * (p3 . 2))) - ((p2 . 2) * ((p1 . 3) * (p3 . 1)))) ; then |{p1,p2,p3}| = (((((p2 . 3) * (p1 . 2)) - ((p2 . 2) * (p1 . 3))) * (p3 . 1)) + ((((p2 . 1) * (p1 . 3)) - ((p2 . 3) * (p1 . 1))) * (p3 . 2))) + ((((p2 . 2) * (p1 . 1)) - ((p2 . 1) * (p1 . 2))) * (p3 . 3)) .= |((p1 p2),|[(p3 . 1),(p3 . 2),(p3 . 3)]|)| by EUCLID_5:30 .= |((p1 p2),p3)| by Th1 ; hence |{p1,p2,p3}| = |((p1 p2),p3)| ; ::_thesis: verum end; theorem :: EUCLID_8:95 for p1, p2, q being Element of REAL 3 holds |{p1,p2,q}| = |((q p1),p2)| proof let p1, p2, q be Element of REAL 3; ::_thesis: |{p1,p2,q}| = |((q p1),p2)| |{p1,p2,q}| = |{p2,q,p1}| by Th93; hence |{p1,p2,q}| = |((q p1),p2)| ; ::_thesis: verum end; begin definition let f1, f2, f3 be PartFunc of REAL,REAL; let t0 be Real; func VFuncdiff (f1,f2,f3,t0) -> Element of REAL 3 equals :: EUCLID_8:def 7 |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]|; coherence |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| is Element of REAL 3 ; end; :: deftheorem defines VFuncdiff EUCLID_8:def_7_:_ for f1, f2, f3 being PartFunc of REAL,REAL for t0 being Real holds VFuncdiff (f1,f2,f3,t0) = |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]|; theorem :: EUCLID_8:96 for f1, f2, f3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 holds VFuncdiff (f1,f2,f3,t0) = (((diff (f1,t0)) * ) + ((diff (f2,t0)) * )) + ((diff (f3,t0)) * ) by Th39; theorem Th97: :: EUCLID_8:97 for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0) = (VFuncdiff (f1,f2,f3,t0)) + (VFuncdiff (g1,g2,g3,t0)) proof let f1, f2, f3, g1, g2, g3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0) = (VFuncdiff (f1,f2,f3,t0)) + (VFuncdiff (g1,g2,g3,t0)) let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 implies VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0) = (VFuncdiff (f1,f2,f3,t0)) + (VFuncdiff (g1,g2,g3,t0)) ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 ) ; ::_thesis: VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0) = (VFuncdiff (f1,f2,f3,t0)) + (VFuncdiff (g1,g2,g3,t0)) set p = |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]|; set q = |[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]|; A3: ( |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 1 = diff (f1,t0) & |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 2 = diff (f2,t0) & |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 3 = diff (f3,t0) ) by FINSEQ_1:45; A4: ( |[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 1 = diff (g1,t0) & |[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 2 = diff (g2,t0) & |[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 3 = diff (g3,t0) ) by FINSEQ_1:45; VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0) = |[((diff (f1,t0)) + (diff (g1,t0))),(diff ((f2 + g2),t0)),(diff ((f3 + g3),t0))]| by A1, A2, FDIFF_1:13 .= |[((diff (f1,t0)) + (diff (g1,t0))),((diff (f2,t0)) + (diff (g2,t0))),(diff ((f3 + g3),t0))]| by A1, A2, FDIFF_1:13 .= |[((|[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 1) + (|[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 1)),((|[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 2) + (|[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 2)),((|[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 3) + (|[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 3))]| by A1, A2, A3, A4, FDIFF_1:13 .= (VFuncdiff (f1,f2,f3,t0)) + (VFuncdiff (g1,g2,g3,t0)) by Lm2 ; hence VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0) = (VFuncdiff (f1,f2,f3,t0)) + (VFuncdiff (g1,g2,g3,t0)) ; ::_thesis: verum end; theorem Th98: :: EUCLID_8:98 for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff ((f1 - g1),(f2 - g2),(f3 - g3),t0) = (VFuncdiff (f1,f2,f3,t0)) - (VFuncdiff (g1,g2,g3,t0)) proof let f1, f2, f3, g1, g2, g3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff ((f1 - g1),(f2 - g2),(f3 - g3),t0) = (VFuncdiff (f1,f2,f3,t0)) - (VFuncdiff (g1,g2,g3,t0)) let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 implies VFuncdiff ((f1 - g1),(f2 - g2),(f3 - g3),t0) = (VFuncdiff (f1,f2,f3,t0)) - (VFuncdiff (g1,g2,g3,t0)) ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 ) ; ::_thesis: VFuncdiff ((f1 - g1),(f2 - g2),(f3 - g3),t0) = (VFuncdiff (f1,f2,f3,t0)) - (VFuncdiff (g1,g2,g3,t0)) set p = |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]|; set q = |[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]|; A3: ( |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 1 = diff (f1,t0) & |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 2 = diff (f2,t0) & |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 3 = diff (f3,t0) ) by FINSEQ_1:45; A4: ( |[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 1 = diff (g1,t0) & |[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 2 = diff (g2,t0) & |[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 3 = diff (g3,t0) ) by FINSEQ_1:45; VFuncdiff ((f1 - g1),(f2 - g2),(f3 - g3),t0) = |[((diff (f1,t0)) - (diff (g1,t0))),(diff ((f2 - g2),t0)),(diff ((f3 - g3),t0))]| by A1, A2, FDIFF_1:14 .= |[((diff (f1,t0)) - (diff (g1,t0))),((diff (f2,t0)) - (diff (g2,t0))),(diff ((f3 - g3),t0))]| by A1, A2, FDIFF_1:14 .= |[((|[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 1) - (|[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 1)),((|[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 2) - (|[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 2)),((|[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 3) - (|[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 3))]| by A1, A2, A3, A4, FDIFF_1:14 .= (VFuncdiff (f1,f2,f3,t0)) - (VFuncdiff (g1,g2,g3,t0)) by Lm4 ; hence VFuncdiff ((f1 - g1),(f2 - g2),(f3 - g3),t0) = (VFuncdiff (f1,f2,f3,t0)) - (VFuncdiff (g1,g2,g3,t0)) ; ::_thesis: verum end; theorem Th99: :: EUCLID_8:99 for r being Element of REAL for f1, f2, f3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 holds VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0) = r * (VFuncdiff (f1,f2,f3,t0)) proof let r be Element of REAL ; ::_thesis: for f1, f2, f3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 holds VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0) = r * (VFuncdiff (f1,f2,f3,t0)) let f1, f2, f3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 holds VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0) = r * (VFuncdiff (f1,f2,f3,t0)) let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 implies VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0) = r * (VFuncdiff (f1,f2,f3,t0)) ) assume A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) ; ::_thesis: VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0) = r * (VFuncdiff (f1,f2,f3,t0)) set p = |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]|; A2: ( |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 1 = diff (f1,t0) & |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 2 = diff (f2,t0) & |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 3 = diff (f3,t0) ) by FINSEQ_1:45; VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0) = |[(r * (diff (f1,t0))),(diff ((r (#) f2),t0)),(diff ((r (#) f3),t0))]| by A1, FDIFF_1:15 .= |[(r * (diff (f1,t0))),(r * (diff (f2,t0))),(diff ((r (#) f3),t0))]| by A1, FDIFF_1:15 .= |[(r * (|[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 1)),(r * (|[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 2)),(r * (|[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 3))]| by A1, A2, FDIFF_1:15 .= r * (VFuncdiff (f1,f2,f3,t0)) by Lm1 ; hence VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0) = r * (VFuncdiff (f1,f2,f3,t0)) ; ::_thesis: verum end; theorem Th100: :: EUCLID_8:100 for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff ((f1 (#) g1),(f2 (#) g2),(f3 (#) g3),t0) = |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]| + |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]| proof let f1, f2, f3, g1, g2, g3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff ((f1 (#) g1),(f2 (#) g2),(f3 (#) g3),t0) = |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]| + |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]| let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 implies VFuncdiff ((f1 (#) g1),(f2 (#) g2),(f3 (#) g3),t0) = |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]| + |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]| ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 ) ; ::_thesis: VFuncdiff ((f1 (#) g1),(f2 (#) g2),(f3 (#) g3),t0) = |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]| + |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]| set p = |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]|; set q = |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]|; A3: ( |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]| . 1 = (g1 . t0) * (diff (f1,t0)) & |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]| . 2 = (g2 . t0) * (diff (f2,t0)) & |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]| . 3 = (g3 . t0) * (diff (f3,t0)) ) by FINSEQ_1:45; A4: ( |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]| . 1 = (f1 . t0) * (diff (g1,t0)) & |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]| . 2 = (f2 . t0) * (diff (g2,t0)) & |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]| . 3 = (f3 . t0) * (diff (g3,t0)) ) by FINSEQ_1:45; VFuncdiff ((f1 (#) g1),(f2 (#) g2),(f3 (#) g3),t0) = |[(((g1 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g1,t0)))),(diff ((f2 (#) g2),t0)),(diff ((f3 (#) g3),t0))]| by A1, A2, FDIFF_1:16 .= |[(((g1 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g1,t0)))),(((g2 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g2,t0)))),(diff ((f3 (#) g3),t0))]| by A1, A2, FDIFF_1:16 .= |[((|[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]| . 1) + (|[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]| . 1)),((|[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]| . 2) + (|[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]| . 2)),((|[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]| . 3) + (|[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]| . 3))]| by A1, A2, A3, A4, FDIFF_1:16 .= |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]| + |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]| by Lm2 ; hence VFuncdiff ((f1 (#) g1),(f2 (#) g2),(f3 (#) g3),t0) = |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]| + |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]| ; ::_thesis: verum end; theorem Th101: :: EUCLID_8:101 for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in f1 . t0 & g2 is_differentiable_in f2 . t0 & g3 is_differentiable_in f3 . t0 holds VFuncdiff ((g1 * f1),(g2 * f2),(g3 * f3),t0) = |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| proof let f1, f2, f3, g1, g2, g3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in f1 . t0 & g2 is_differentiable_in f2 . t0 & g3 is_differentiable_in f3 . t0 holds VFuncdiff ((g1 * f1),(g2 * f2),(g3 * f3),t0) = |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in f1 . t0 & g2 is_differentiable_in f2 . t0 & g3 is_differentiable_in f3 . t0 implies VFuncdiff ((g1 * f1),(g2 * f2),(g3 * f3),t0) = |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( g1 is_differentiable_in f1 . t0 & g2 is_differentiable_in f2 . t0 & g3 is_differentiable_in f3 . t0 ) ; ::_thesis: VFuncdiff ((g1 * f1),(g2 * f2),(g3 * f3),t0) = |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| VFuncdiff ((g1 * f1),(g2 * f2),(g3 * f3),t0) = |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),(diff ((g2 * f2),t0)),(diff ((g3 * f3),t0))]| by A1, A2, FDIFF_2:13 .= |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),(diff ((g3 * f3),t0))]| by A1, A2, FDIFF_2:13 .= |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| by A1, A2, FDIFF_2:13 ; hence VFuncdiff ((g1 * f1),(g2 * f2),(g3 * f3),t0) = |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| ; ::_thesis: verum end; theorem :: EUCLID_8:102 for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & g1 . t0 <> 0 & g2 . t0 <> 0 & g3 . t0 <> 0 holds VFuncdiff ((f1 / g1),(f2 / g2),(f3 / g3),t0) = |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2)),((((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2))]| proof let f1, f2, f3, g1, g2, g3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & g1 . t0 <> 0 & g2 . t0 <> 0 & g3 . t0 <> 0 holds VFuncdiff ((f1 / g1),(f2 / g2),(f3 / g3),t0) = |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2)),((((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2))]| let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & g1 . t0 <> 0 & g2 . t0 <> 0 & g3 . t0 <> 0 implies VFuncdiff ((f1 / g1),(f2 / g2),(f3 / g3),t0) = |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2)),((((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2))]| ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 ) and A3: ( g1 . t0 <> 0 & g2 . t0 <> 0 & g3 . t0 <> 0 ) ; ::_thesis: VFuncdiff ((f1 / g1),(f2 / g2),(f3 / g3),t0) = |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2)),((((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2))]| VFuncdiff ((f1 / g1),(f2 / g2),(f3 / g3),t0) = |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),(diff ((f2 / g2),t0)),(diff ((f3 / g3),t0))]| by A1, A2, A3, FDIFF_2:14 .= |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2)),(diff ((f3 / g3),t0))]| by A1, A2, A3, FDIFF_2:14 .= |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2)),((((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2))]| by A1, A2, A3, FDIFF_2:14 ; hence VFuncdiff ((f1 / g1),(f2 / g2),(f3 / g3),t0) = |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2)),((((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2))]| ; ::_thesis: verum end; theorem Th103: :: EUCLID_8:103 for f1, f2, f3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & f1 . t0 <> 0 & f2 . t0 <> 0 & f3 . t0 <> 0 holds VFuncdiff ((f1 ^),(f2 ^),(f3 ^),t0) = - |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]| proof let f1, f2, f3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & f1 . t0 <> 0 & f2 . t0 <> 0 & f3 . t0 <> 0 holds VFuncdiff ((f1 ^),(f2 ^),(f3 ^),t0) = - |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]| let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & f1 . t0 <> 0 & f2 . t0 <> 0 & f3 . t0 <> 0 implies VFuncdiff ((f1 ^),(f2 ^),(f3 ^),t0) = - |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]| ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( f1 . t0 <> 0 & f2 . t0 <> 0 & f3 . t0 <> 0 ) ; ::_thesis: VFuncdiff ((f1 ^),(f2 ^),(f3 ^),t0) = - |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]| set p = |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]|; A3: ( |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]| . 1 = (diff (f1,t0)) / ((f1 . t0) ^2) & |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]| . 2 = (diff (f2,t0)) / ((f2 . t0) ^2) & |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]| . 3 = (diff (f3,t0)) / ((f3 . t0) ^2) ) by FINSEQ_1:45; VFuncdiff ((f1 ^),(f2 ^),(f3 ^),t0) = |[(- ((diff (f1,t0)) / ((f1 . t0) ^2))),(diff ((f2 ^),t0)),(diff ((f3 ^),t0))]| by A1, A2, FDIFF_2:15 .= |[(- ((diff (f1,t0)) / ((f1 . t0) ^2))),(- ((diff (f2,t0)) / ((f2 . t0) ^2))),(diff ((f3 ^),t0))]| by A1, A2, FDIFF_2:15 .= |[(- ((diff (f1,t0)) / ((f1 . t0) ^2))),(- ((diff (f2,t0)) / ((f2 . t0) ^2))),(- ((diff (f3,t0)) / ((f3 . t0) ^2)))]| by A1, A2, FDIFF_2:15 .= |[((- 1) * (|[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]| . 1)),((- 1) * (|[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]| . 2)),((- 1) * (|[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]| . 3))]| by A3 .= - |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]| by Lm1 ; hence VFuncdiff ((f1 ^),(f2 ^),(f3 ^),t0) = - |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]| ; ::_thesis: verum end; theorem :: EUCLID_8:104 for r being Element of REAL for f1, f2, f3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 holds VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0) = (((r * (diff (f1,t0))) * ) + ((r * (diff (f2,t0))) * )) + ((r * (diff (f3,t0))) * ) proof let r be Element of REAL ; ::_thesis: for f1, f2, f3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 holds VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0) = (((r * (diff (f1,t0))) * ) + ((r * (diff (f2,t0))) * )) + ((r * (diff (f3,t0))) * ) let f1, f2, f3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 holds VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0) = (((r * (diff (f1,t0))) * ) + ((r * (diff (f2,t0))) * )) + ((r * (diff (f3,t0))) * ) let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 implies VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0) = (((r * (diff (f1,t0))) * ) + ((r * (diff (f2,t0))) * )) + ((r * (diff (f3,t0))) * ) ) assume ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) ; ::_thesis: VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0) = (((r * (diff (f1,t0))) * ) + ((r * (diff (f2,t0))) * )) + ((r * (diff (f3,t0))) * ) then VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0) = r * (VFuncdiff (f1,f2,f3,t0)) by Th99 .= r * ((((diff (f1,t0)) * ) + ((diff (f2,t0)) * )) + ((diff (f3,t0)) * )) by Th39 .= r * ((|[(diff (f1,t0)),0,0]| + ((diff (f2,t0)) * )) + ((diff (f3,t0)) * )) by Th25 .= r * ((|[(diff (f1,t0)),0,0]| + |[0,(diff (f2,t0)),0]|) + ((diff (f3,t0)) * )) by Th26 .= r * ((|[(diff (f1,t0)),0,0]| + |[0,(diff (f2,t0)),0]|) + |[0,0,(diff (f3,t0))]|) by Th27 .= r * (|[((diff (f1,t0)) + 0),(0 + (diff (f2,t0))),(0 + 0)]| + |[0,0,(diff (f3,t0))]|) by Lm8 .= r * |[((diff (f1,t0)) + 0),((diff (f2,t0)) + 0),(0 + (diff (f3,t0)))]| by Lm8 .= (((r * (diff (f1,t0))) * ) + ((r * (diff (f2,t0))) * )) + ((r * (diff (f3,t0))) * ) by Th40 ; hence VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0) = (((r * (diff (f1,t0))) * ) + ((r * (diff (f2,t0))) * )) + ((r * (diff (f3,t0))) * ) ; ::_thesis: verum end; theorem :: EUCLID_8:105 for r being Element of REAL for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff ((r (#) (f1 + g1)),(r (#) (f2 + g2)),(r (#) (f3 + g3)),t0) = (r * (VFuncdiff (f1,f2,f3,t0))) + (r * (VFuncdiff (g1,g2,g3,t0))) proof let r be Element of REAL ; ::_thesis: for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff ((r (#) (f1 + g1)),(r (#) (f2 + g2)),(r (#) (f3 + g3)),t0) = (r * (VFuncdiff (f1,f2,f3,t0))) + (r * (VFuncdiff (g1,g2,g3,t0))) let f1, f2, f3, g1, g2, g3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff ((r (#) (f1 + g1)),(r (#) (f2 + g2)),(r (#) (f3 + g3)),t0) = (r * (VFuncdiff (f1,f2,f3,t0))) + (r * (VFuncdiff (g1,g2,g3,t0))) let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 implies VFuncdiff ((r (#) (f1 + g1)),(r (#) (f2 + g2)),(r (#) (f3 + g3)),t0) = (r * (VFuncdiff (f1,f2,f3,t0))) + (r * (VFuncdiff (g1,g2,g3,t0))) ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 ) ; ::_thesis: VFuncdiff ((r (#) (f1 + g1)),(r (#) (f2 + g2)),(r (#) (f3 + g3)),t0) = (r * (VFuncdiff (f1,f2,f3,t0))) + (r * (VFuncdiff (g1,g2,g3,t0))) ( f1 + g1 is_differentiable_in t0 & f2 + g2 is_differentiable_in t0 & f3 + g3 is_differentiable_in t0 ) by A1, A2, FDIFF_1:13; then VFuncdiff ((r (#) (f1 + g1)),(r (#) (f2 + g2)),(r (#) (f3 + g3)),t0) = r * (VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0)) by Th99 .= r * ((VFuncdiff (f1,f2,f3,t0)) + (VFuncdiff (g1,g2,g3,t0))) by A1, A2, Th97 .= (r * (VFuncdiff (f1,f2,f3,t0))) + (r * (VFuncdiff (g1,g2,g3,t0))) by EUCLID_4:6 ; hence VFuncdiff ((r (#) (f1 + g1)),(r (#) (f2 + g2)),(r (#) (f3 + g3)),t0) = (r * (VFuncdiff (f1,f2,f3,t0))) + (r * (VFuncdiff (g1,g2,g3,t0))) ; ::_thesis: verum end; theorem :: EUCLID_8:106 for r being Element of REAL for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff ((r (#) (f1 - g1)),(r (#) (f2 - g2)),(r (#) (f3 - g3)),t0) = (r * (VFuncdiff (f1,f2,f3,t0))) - (r * (VFuncdiff (g1,g2,g3,t0))) proof let r be Element of REAL ; ::_thesis: for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff ((r (#) (f1 - g1)),(r (#) (f2 - g2)),(r (#) (f3 - g3)),t0) = (r * (VFuncdiff (f1,f2,f3,t0))) - (r * (VFuncdiff (g1,g2,g3,t0))) let f1, f2, f3, g1, g2, g3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff ((r (#) (f1 - g1)),(r (#) (f2 - g2)),(r (#) (f3 - g3)),t0) = (r * (VFuncdiff (f1,f2,f3,t0))) - (r * (VFuncdiff (g1,g2,g3,t0))) let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 implies VFuncdiff ((r (#) (f1 - g1)),(r (#) (f2 - g2)),(r (#) (f3 - g3)),t0) = (r * (VFuncdiff (f1,f2,f3,t0))) - (r * (VFuncdiff (g1,g2,g3,t0))) ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 ) ; ::_thesis: VFuncdiff ((r (#) (f1 - g1)),(r (#) (f2 - g2)),(r (#) (f3 - g3)),t0) = (r * (VFuncdiff (f1,f2,f3,t0))) - (r * (VFuncdiff (g1,g2,g3,t0))) ( f1 - g1 is_differentiable_in t0 & f2 - g2 is_differentiable_in t0 & f3 - g3 is_differentiable_in t0 ) by A1, A2, FDIFF_1:14; then VFuncdiff ((r (#) (f1 - g1)),(r (#) (f2 - g2)),(r (#) (f3 - g3)),t0) = r * (VFuncdiff ((f1 - g1),(f2 - g2),(f3 - g3),t0)) by Th99 .= r * ((VFuncdiff (f1,f2,f3,t0)) - (VFuncdiff (g1,g2,g3,t0))) by A1, A2, Th98 .= (r * (VFuncdiff (f1,f2,f3,t0))) - (r * (VFuncdiff (g1,g2,g3,t0))) by EUCLIDLP:12 ; hence VFuncdiff ((r (#) (f1 - g1)),(r (#) (f2 - g2)),(r (#) (f3 - g3)),t0) = (r * (VFuncdiff (f1,f2,f3,t0))) - (r * (VFuncdiff (g1,g2,g3,t0))) ; ::_thesis: verum end; theorem :: EUCLID_8:107 for r being Element of REAL for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff (((r (#) f1) (#) g1),((r (#) f2) (#) g2),((r (#) f3) (#) g3),t0) = (r * |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]|) + (r * |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]|) proof let r be Element of REAL ; ::_thesis: for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff (((r (#) f1) (#) g1),((r (#) f2) (#) g2),((r (#) f3) (#) g3),t0) = (r * |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]|) + (r * |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]|) let f1, f2, f3, g1, g2, g3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff (((r (#) f1) (#) g1),((r (#) f2) (#) g2),((r (#) f3) (#) g3),t0) = (r * |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]|) + (r * |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]|) let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 implies VFuncdiff (((r (#) f1) (#) g1),((r (#) f2) (#) g2),((r (#) f3) (#) g3),t0) = (r * |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]|) + (r * |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]|) ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 ) ; ::_thesis: VFuncdiff (((r (#) f1) (#) g1),((r (#) f2) (#) g2),((r (#) f3) (#) g3),t0) = (r * |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]|) + (r * |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]|) ( r (#) f1 is_differentiable_in t0 & r (#) f2 is_differentiable_in t0 & r (#) f3 is_differentiable_in t0 ) by A1, FDIFF_1:15; then VFuncdiff (((r (#) f1) (#) g1),((r (#) f2) (#) g2),((r (#) f3) (#) g3),t0) = |[((g1 . t0) * (diff ((r (#) f1),t0))),((g2 . t0) * (diff ((r (#) f2),t0))),((g3 . t0) * (diff ((r (#) f3),t0)))]| + |[(((r (#) f1) . t0) * (diff (g1,t0))),(((r (#) f2) . t0) * (diff (g2,t0))),(((r (#) f3) . t0) * (diff (g3,t0)))]| by A2, Th100 .= |[((g1 . t0) * (r * (diff (f1,t0)))),((g2 . t0) * (diff ((r (#) f2),t0))),((g3 . t0) * (diff ((r (#) f3),t0)))]| + |[(((r (#) f1) . t0) * (diff (g1,t0))),(((r (#) f2) . t0) * (diff (g2,t0))),(((r (#) f3) . t0) * (diff (g3,t0)))]| by A1, FDIFF_1:15 .= |[((g1 . t0) * (r * (diff (f1,t0)))),((g2 . t0) * (r * (diff (f2,t0)))),((g3 . t0) * (diff ((r (#) f3),t0)))]| + |[(((r (#) f1) . t0) * (diff (g1,t0))),(((r (#) f2) . t0) * (diff (g2,t0))),(((r (#) f3) . t0) * (diff (g3,t0)))]| by A1, FDIFF_1:15 .= |[((g1 . t0) * (r * (diff (f1,t0)))),((g2 . t0) * (r * (diff (f2,t0)))),((g3 . t0) * (r * (diff (f3,t0))))]| + |[(((r (#) f1) . t0) * (diff (g1,t0))),(((r (#) f2) . t0) * (diff (g2,t0))),(((r (#) f3) . t0) * (diff (g3,t0)))]| by A1, FDIFF_1:15 .= |[((g1 . t0) * (r * (diff (f1,t0)))),((g2 . t0) * (r * (diff (f2,t0)))),((g3 . t0) * (r * (diff (f3,t0))))]| + |[((r * (f1 . t0)) * (diff (g1,t0))),(((r (#) f2) . t0) * (diff (g2,t0))),(((r (#) f3) . t0) * (diff (g3,t0)))]| by VALUED_1:6 .= |[((g1 . t0) * (r * (diff (f1,t0)))),((g2 . t0) * (r * (diff (f2,t0)))),((g3 . t0) * (r * (diff (f3,t0))))]| + |[((r * (f1 . t0)) * (diff (g1,t0))),((r * (f2 . t0)) * (diff (g2,t0))),(((r (#) f3) . t0) * (diff (g3,t0)))]| by VALUED_1:6 .= |[((g1 . t0) * (r * (diff (f1,t0)))),((g2 . t0) * (r * (diff (f2,t0)))),((g3 . t0) * (r * (diff (f3,t0))))]| + |[((r * (f1 . t0)) * (diff (g1,t0))),((r * (f2 . t0)) * (diff (g2,t0))),((r * (f3 . t0)) * (diff (g3,t0)))]| by VALUED_1:6 .= |[((g1 . t0) * (r * (diff (f1,t0)))),((g2 . t0) * (r * (diff (f2,t0)))),((g3 . t0) * (r * (diff (f3,t0))))]| + |[(r * ((f1 . t0) * (diff (g1,t0)))),(r * ((f2 . t0) * (diff (g2,t0)))),(r * ((f3 . t0) * (diff (g3,t0))))]| .= |[(r * ((g1 . t0) * (diff (f1,t0)))),(r * ((g2 . t0) * (diff (f2,t0)))),(r * ((g3 . t0) * (diff (f3,t0))))]| + (r * |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]|) by Lm6 .= (r * |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]|) + (r * |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]|) by Lm6 ; hence VFuncdiff (((r (#) f1) (#) g1),((r (#) f2) (#) g2),((r (#) f3) (#) g3),t0) = (r * |[((g1 . t0) * (diff (f1,t0))),((g2 . t0) * (diff (f2,t0))),((g3 . t0) * (diff (f3,t0)))]|) + (r * |[((f1 . t0) * (diff (g1,t0))),((f2 . t0) * (diff (g2,t0))),((f3 . t0) * (diff (g3,t0)))]|) ; ::_thesis: verum end; theorem :: EUCLID_8:108 for r being Element of REAL for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in f1 . t0 & g2 is_differentiable_in f2 . t0 & g3 is_differentiable_in f3 . t0 holds VFuncdiff (((r (#) g1) * f1),((r (#) g2) * f2),((r (#) g3) * f3),t0) = r * |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| proof let r be Element of REAL ; ::_thesis: for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in f1 . t0 & g2 is_differentiable_in f2 . t0 & g3 is_differentiable_in f3 . t0 holds VFuncdiff (((r (#) g1) * f1),((r (#) g2) * f2),((r (#) g3) * f3),t0) = r * |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| let f1, f2, f3, g1, g2, g3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in f1 . t0 & g2 is_differentiable_in f2 . t0 & g3 is_differentiable_in f3 . t0 holds VFuncdiff (((r (#) g1) * f1),((r (#) g2) * f2),((r (#) g3) * f3),t0) = r * |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in f1 . t0 & g2 is_differentiable_in f2 . t0 & g3 is_differentiable_in f3 . t0 implies VFuncdiff (((r (#) g1) * f1),((r (#) g2) * f2),((r (#) g3) * f3),t0) = r * |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( g1 is_differentiable_in f1 . t0 & g2 is_differentiable_in f2 . t0 & g3 is_differentiable_in f3 . t0 ) ; ::_thesis: VFuncdiff (((r (#) g1) * f1),((r (#) g2) * f2),((r (#) g3) * f3),t0) = r * |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| ( r (#) g1 is_differentiable_in f1 . t0 & r (#) g2 is_differentiable_in f2 . t0 & r (#) g3 is_differentiable_in f3 . t0 ) by A2, FDIFF_1:15; then VFuncdiff (((r (#) g1) * f1),((r (#) g2) * f2),((r (#) g3) * f3),t0) = |[((diff ((r (#) g1),(f1 . t0))) * (diff (f1,t0))),((diff ((r (#) g2),(f2 . t0))) * (diff (f2,t0))),((diff ((r (#) g3),(f3 . t0))) * (diff (f3,t0)))]| by A1, Th101 .= |[((r * (diff (g1,(f1 . t0)))) * (diff (f1,t0))),((diff ((r (#) g2),(f2 . t0))) * (diff (f2,t0))),((diff ((r (#) g3),(f3 . t0))) * (diff (f3,t0)))]| by A2, FDIFF_1:15 .= |[((r * (diff (g1,(f1 . t0)))) * (diff (f1,t0))),((r * (diff (g2,(f2 . t0)))) * (diff (f2,t0))),((diff ((r (#) g3),(f3 . t0))) * (diff (f3,t0)))]| by A2, FDIFF_1:15 .= |[((r * (diff (g1,(f1 . t0)))) * (diff (f1,t0))),((r * (diff (g2,(f2 . t0)))) * (diff (f2,t0))),((r * (diff (g3,(f3 . t0)))) * (diff (f3,t0)))]| by A2, FDIFF_1:15 .= |[(r * ((diff (g1,(f1 . t0))) * (diff (f1,t0)))),(r * ((diff (g2,(f2 . t0))) * (diff (f2,t0)))),(r * ((diff (g3,(f3 . t0))) * (diff (f3,t0))))]| .= r * |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| by Lm6 ; hence VFuncdiff (((r (#) g1) * f1),((r (#) g2) * f2),((r (#) g3) * f3),t0) = r * |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| ; ::_thesis: verum end; theorem :: EUCLID_8:109 for r being Element of REAL for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & g1 . t0 <> 0 & g2 . t0 <> 0 & g3 . t0 <> 0 holds VFuncdiff (((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0) = r * |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2)),((((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2))]| proof let r be Element of REAL ; ::_thesis: for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & g1 . t0 <> 0 & g2 . t0 <> 0 & g3 . t0 <> 0 holds VFuncdiff (((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0) = r * |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2)),((((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2))]| let f1, f2, f3, g1, g2, g3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & g1 . t0 <> 0 & g2 . t0 <> 0 & g3 . t0 <> 0 holds VFuncdiff (((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0) = r * |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2)),((((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2))]| let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & g1 . t0 <> 0 & g2 . t0 <> 0 & g3 . t0 <> 0 implies VFuncdiff (((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0) = r * |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2)),((((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2))]| ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 ) and A3: ( g1 . t0 <> 0 & g2 . t0 <> 0 & g3 . t0 <> 0 ) ; ::_thesis: VFuncdiff (((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0) = r * |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2)),((((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2))]| A4: ( r (#) f1 is_differentiable_in t0 & r (#) f2 is_differentiable_in t0 & r (#) f3 is_differentiable_in t0 ) by A1, FDIFF_1:15; then VFuncdiff (((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0) = |[((((diff ((r (#) f1),t0)) * (g1 . t0)) - ((diff (g1,t0)) * ((r (#) f1) . t0))) / ((g1 . t0) ^2)),(diff (((r (#) f2) / g2),t0)),(diff (((r (#) f3) / g3),t0))]| by A2, A3, FDIFF_2:14 .= |[((((diff ((r (#) f1),t0)) * (g1 . t0)) - ((diff (g1,t0)) * ((r (#) f1) . t0))) / ((g1 . t0) ^2)),((((diff ((r (#) f2),t0)) * (g2 . t0)) - ((diff (g2,t0)) * ((r (#) f2) . t0))) / ((g2 . t0) ^2)),(diff (((r (#) f3) / g3),t0))]| by A2, A3, A4, FDIFF_2:14 .= |[((((diff ((r (#) f1),t0)) * (g1 . t0)) - ((diff (g1,t0)) * ((r (#) f1) . t0))) / ((g1 . t0) ^2)),((((diff ((r (#) f2),t0)) * (g2 . t0)) - ((diff (g2,t0)) * ((r (#) f2) . t0))) / ((g2 . t0) ^2)),((((diff ((r (#) f3),t0)) * (g3 . t0)) - ((diff (g3,t0)) * ((r (#) f3) . t0))) / ((g3 . t0) ^2))]| by A2, A3, A4, FDIFF_2:14 .= |[((((r * (diff (f1,t0))) * (g1 . t0)) - ((diff (g1,t0)) * ((r (#) f1) . t0))) / ((g1 . t0) ^2)),((((diff ((r (#) f2),t0)) * (g2 . t0)) - ((diff (g2,t0)) * ((r (#) f2) . t0))) / ((g2 . t0) ^2)),((((diff ((r (#) f3),t0)) * (g3 . t0)) - ((diff (g3,t0)) * ((r (#) f3) . t0))) / ((g3 . t0) ^2))]| by A1, FDIFF_1:15 .= |[((((r * (diff (f1,t0))) * (g1 . t0)) - ((diff (g1,t0)) * ((r (#) f1) . t0))) / ((g1 . t0) ^2)),((((r * (diff (f2,t0))) * (g2 . t0)) - ((diff (g2,t0)) * ((r (#) f2) . t0))) / ((g2 . t0) ^2)),((((diff ((r (#) f3),t0)) * (g3 . t0)) - ((diff (g3,t0)) * ((r (#) f3) . t0))) / ((g3 . t0) ^2))]| by A1, FDIFF_1:15 .= |[((((r * (diff (f1,t0))) * (g1 . t0)) - ((diff (g1,t0)) * ((r (#) f1) . t0))) / ((g1 . t0) ^2)),((((r * (diff (f2,t0))) * (g2 . t0)) - ((diff (g2,t0)) * ((r (#) f2) . t0))) / ((g2 . t0) ^2)),((((r * (diff (f3,t0))) * (g3 . t0)) - ((diff (g3,t0)) * ((r (#) f3) . t0))) / ((g3 . t0) ^2))]| by A1, FDIFF_1:15 .= |[((((r * (diff (f1,t0))) * (g1 . t0)) - ((diff (g1,t0)) * (r * (f1 . t0)))) / ((g1 . t0) ^2)),((((r * (diff (f2,t0))) * (g2 . t0)) - ((diff (g2,t0)) * ((r (#) f2) . t0))) / ((g2 . t0) ^2)),((((r * (diff (f3,t0))) * (g3 . t0)) - ((diff (g3,t0)) * ((r (#) f3) . t0))) / ((g3 . t0) ^2))]| by VALUED_1:6 .= |[((((r * (diff (f1,t0))) * (g1 . t0)) - ((diff (g1,t0)) * (r * (f1 . t0)))) / ((g1 . t0) ^2)),((((r * (diff (f2,t0))) * (g2 . t0)) - ((diff (g2,t0)) * (r * (f2 . t0)))) / ((g2 . t0) ^2)),((((r * (diff (f3,t0))) * (g3 . t0)) - ((diff (g3,t0)) * ((r (#) f3) . t0))) / ((g3 . t0) ^2))]| by VALUED_1:6 .= |[((((r * (diff (f1,t0))) * (g1 . t0)) - ((diff (g1,t0)) * (r * (f1 . t0)))) / ((g1 . t0) ^2)),((((r * (diff (f2,t0))) * (g2 . t0)) - ((diff (g2,t0)) * (r * (f2 . t0)))) / ((g2 . t0) ^2)),((((r * (diff (f3,t0))) * (g3 . t0)) - ((diff (g3,t0)) * (r * (f3 . t0)))) / ((g3 . t0) ^2))]| by VALUED_1:6 .= |[((r * (((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0)))) / ((g1 . t0) ^2)),((r * (((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0)))) / ((g2 . t0) ^2)),((r * (((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0)))) / ((g3 . t0) ^2))]| .= |[(r * ((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2))),((r * (((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0)))) / ((g2 . t0) ^2)),((r * (((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0)))) / ((g3 . t0) ^2))]| by XCMPLX_1:74 .= |[(r * ((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2))),(r * ((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2))),((r * (((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0)))) / ((g3 . t0) ^2))]| by XCMPLX_1:74 .= |[(r * ((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2))),(r * ((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2))),(r * ((((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2)))]| by XCMPLX_1:74 .= r * |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2)),((((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2))]| by Lm6 ; hence VFuncdiff (((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0) = r * |[((((diff (f1,t0)) * (g1 . t0)) - ((diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2)),((((diff (f2,t0)) * (g2 . t0)) - ((diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2)),((((diff (f3,t0)) * (g3 . t0)) - ((diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2))]| ; ::_thesis: verum end; theorem :: EUCLID_8:110 for r being Element of REAL for f1, f2, f3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & f1 . t0 <> 0 & f2 . t0 <> 0 & f3 . t0 <> 0 & r <> 0 holds VFuncdiff (((r (#) f1) ^),((r (#) f2) ^),((r (#) f3) ^),t0) = - ((1 / r) * |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]|) proof let r be Element of REAL ; ::_thesis: for f1, f2, f3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & f1 . t0 <> 0 & f2 . t0 <> 0 & f3 . t0 <> 0 & r <> 0 holds VFuncdiff (((r (#) f1) ^),((r (#) f2) ^),((r (#) f3) ^),t0) = - ((1 / r) * |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]|) let f1, f2, f3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & f1 . t0 <> 0 & f2 . t0 <> 0 & f3 . t0 <> 0 & r <> 0 holds VFuncdiff (((r (#) f1) ^),((r (#) f2) ^),((r (#) f3) ^),t0) = - ((1 / r) * |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]|) let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & f1 . t0 <> 0 & f2 . t0 <> 0 & f3 . t0 <> 0 & r <> 0 implies VFuncdiff (((r (#) f1) ^),((r (#) f2) ^),((r (#) f3) ^),t0) = - ((1 / r) * |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]|) ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( f1 . t0 <> 0 & f2 . t0 <> 0 & f3 . t0 <> 0 ) and A3: r <> 0 ; ::_thesis: VFuncdiff (((r (#) f1) ^),((r (#) f2) ^),((r (#) f3) ^),t0) = - ((1 / r) * |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]|) A4: ( r (#) f1 is_differentiable_in t0 & r (#) f2 is_differentiable_in t0 & r (#) f3 is_differentiable_in t0 ) by A1, FDIFF_1:15; A5: (r (#) f1) . t0 = r * (f1 . t0) by VALUED_1:6; A6: (r (#) f2) . t0 = r * (f2 . t0) by VALUED_1:6; A7: (r (#) f3) . t0 = r * (f3 . t0) by VALUED_1:6; then VFuncdiff (((r (#) f1) ^),((r (#) f2) ^),((r (#) f3) ^),t0) = - |[((diff ((r (#) f1),t0)) / (((r (#) f1) . t0) ^2)),((diff ((r (#) f2),t0)) / (((r (#) f2) . t0) ^2)),((diff ((r (#) f3),t0)) / (((r (#) f3) . t0) ^2))]| by A4, A5, A6, A2, A3, Th103 .= - |[((r * (diff (f1,t0))) / (((r (#) f1) . t0) ^2)),((diff ((r (#) f2),t0)) / (((r (#) f2) . t0) ^2)),((diff ((r (#) f3),t0)) / (((r (#) f3) . t0) ^2))]| by A1, FDIFF_1:15 .= - |[((r * (diff (f1,t0))) / (((r (#) f1) . t0) ^2)),((r * (diff (f2,t0))) / (((r (#) f2) . t0) ^2)),((diff ((r (#) f3),t0)) / (((r (#) f3) . t0) ^2))]| by A1, FDIFF_1:15 .= - |[((r * (diff (f1,t0))) / (((r (#) f1) . t0) ^2)),((r * (diff (f2,t0))) / (((r (#) f2) . t0) ^2)),((r * (diff (f3,t0))) / (((r (#) f3) . t0) ^2))]| by A1, FDIFF_1:15 .= - |[(r * ((diff (f1,t0)) / (((r (#) f1) . t0) ^2))),((r * (diff (f2,t0))) / (((r (#) f2) . t0) ^2)),((r * (diff (f3,t0))) / (((r (#) f3) . t0) ^2))]| by XCMPLX_1:74 .= - |[(r * ((diff (f1,t0)) / (((r (#) f1) . t0) ^2))),(r * ((diff (f2,t0)) / (((r (#) f2) . t0) ^2))),((r * (diff (f3,t0))) / (((r (#) f3) . t0) ^2))]| by XCMPLX_1:74 .= - |[(r * ((diff (f1,t0)) / (((r (#) f1) . t0) ^2))),(r * ((diff (f2,t0)) / (((r (#) f2) . t0) ^2))),(r * ((diff (f3,t0)) / (((r (#) f3) . t0) ^2)))]| by XCMPLX_1:74 .= - (r * |[((diff (f1,t0)) / ((r ^2) * ((f1 . t0) ^2))),((diff (f2,t0)) / ((r ^2) * ((f2 . t0) ^2))),((diff (f3,t0)) / ((r ^2) * ((f3 . t0) ^2)))]|) by A7, A6, A5, Lm6 .= - (r * |[(((diff (f1,t0)) / (r ^2)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((r ^2) * ((f2 . t0) ^2))),((diff (f3,t0)) / ((r ^2) * ((f3 . t0) ^2)))]|) by XCMPLX_1:78 .= - (r * |[(((diff (f1,t0)) / (r ^2)) / ((f1 . t0) ^2)),(((diff (f2,t0)) / (r ^2)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((r ^2) * ((f3 . t0) ^2)))]|) by XCMPLX_1:78 .= - (r * |[(((diff (f1,t0)) / (r ^2)) / ((f1 . t0) ^2)),(((diff (f2,t0)) / (r ^2)) / ((f2 . t0) ^2)),(((diff (f3,t0)) / (r ^2)) / ((f3 . t0) ^2))]|) by XCMPLX_1:78 .= - (r * |[(((diff (f1,t0)) / ((f1 . t0) ^2)) / (r ^2)),(((diff (f2,t0)) / (r ^2)) / ((f2 . t0) ^2)),(((diff (f3,t0)) / (r ^2)) / ((f3 . t0) ^2))]|) by XCMPLX_1:48 .= - (r * |[(((diff (f1,t0)) / ((f1 . t0) ^2)) / (r ^2)),(((diff (f2,t0)) / ((f2 . t0) ^2)) / (r ^2)),(((diff (f3,t0)) / (r ^2)) / ((f3 . t0) ^2))]|) by XCMPLX_1:48 .= - (r * |[(((diff (f1,t0)) / ((f1 . t0) ^2)) / (r ^2)),(((diff (f2,t0)) / ((f2 . t0) ^2)) / (r ^2)),(((diff (f3,t0)) / ((f3 . t0) ^2)) / (r ^2))]|) by XCMPLX_1:48 .= - (r * |[(((diff (f1,t0)) / ((f1 . t0) ^2)) / ((1 / (r ^2)) ")),(((diff (f2,t0)) / ((f2 . t0) ^2)) / (r ^2)),(((diff (f3,t0)) / ((f3 . t0) ^2)) / (r ^2))]|) by XCMPLX_1:217 .= - (r * |[((1 / (r ^2)) * ((diff (f1,t0)) / ((f1 . t0) ^2))),(((diff (f2,t0)) / ((f2 . t0) ^2)) / (r ^2)),(((diff (f3,t0)) / ((f3 . t0) ^2)) / (r ^2))]|) by XCMPLX_1:219 .= - (r * |[((1 / (r ^2)) * ((diff (f1,t0)) / ((f1 . t0) ^2))),(((diff (f2,t0)) / ((f2 . t0) ^2)) / ((1 / (r ^2)) ")),(((diff (f3,t0)) / ((f3 . t0) ^2)) / (r ^2))]|) by XCMPLX_1:217 .= - (r * |[((1 / (r ^2)) * ((diff (f1,t0)) / ((f1 . t0) ^2))),((1 / (r ^2)) * ((diff (f2,t0)) / ((f2 . t0) ^2))),(((diff (f3,t0)) / ((f3 . t0) ^2)) / (r ^2))]|) by XCMPLX_1:219 .= - (r * |[((1 / (r ^2)) * ((diff (f1,t0)) / ((f1 . t0) ^2))),((1 / (r ^2)) * ((diff (f2,t0)) / ((f2 . t0) ^2))),(((diff (f3,t0)) / ((f3 . t0) ^2)) / ((1 / (r ^2)) "))]|) by XCMPLX_1:217 .= - (r * |[((1 / (r ^2)) * ((diff (f1,t0)) / ((f1 . t0) ^2))),((1 / (r ^2)) * ((diff (f2,t0)) / ((f2 . t0) ^2))),((1 / (r ^2)) * ((diff (f3,t0)) / ((f3 . t0) ^2)))]|) by XCMPLX_1:219 .= - (r * ((1 / (r ^2)) * |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]|)) by Lm6 .= - ((r * (1 / (r ^2))) * |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]|) by EUCLID_4:4 .= - (((r * 1) / (r ^2)) * |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]|) by XCMPLX_1:74 .= - (((r / r) / r) * |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]|) by XCMPLX_1:78 .= - ((1 / r) * |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]|) by A3, XCMPLX_1:60 ; hence VFuncdiff (((r (#) f1) ^),((r (#) f2) ^),((r (#) f3) ^),t0) = - ((1 / r) * |[((diff (f1,t0)) / ((f1 . t0) ^2)),((diff (f2,t0)) / ((f2 . t0) ^2)),((diff (f3,t0)) / ((f3 . t0) ^2))]|) ; ::_thesis: verum end; theorem :: EUCLID_8:111 for f1, f2, f3, g1, g2, g3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff (((f2 (#) g3) - (f3 (#) g2)),((f3 (#) g1) - (f1 (#) g3)),((f1 (#) g2) - (f2 (#) g1)),t0) = |[(((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0)))),(((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0)))),(((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))]| + |[(((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0))),(((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))),(((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))]| proof let f1, f2, f3, g1, g2, g3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds VFuncdiff (((f2 (#) g3) - (f3 (#) g2)),((f3 (#) g1) - (f1 (#) g3)),((f1 (#) g2) - (f2 (#) g1)),t0) = |[(((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0)))),(((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0)))),(((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))]| + |[(((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0))),(((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))),(((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))]| let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 implies VFuncdiff (((f2 (#) g3) - (f3 (#) g2)),((f3 (#) g1) - (f1 (#) g3)),((f1 (#) g2) - (f2 (#) g1)),t0) = |[(((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0)))),(((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0)))),(((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))]| + |[(((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0))),(((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))),(((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))]| ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 ) ; ::_thesis: VFuncdiff (((f2 (#) g3) - (f3 (#) g2)),((f3 (#) g1) - (f1 (#) g3)),((f1 (#) g2) - (f2 (#) g1)),t0) = |[(((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0)))),(((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0)))),(((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))]| + |[(((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0))),(((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))),(((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))]| ( f2 (#) g3 is_differentiable_in t0 & f3 (#) g2 is_differentiable_in t0 & f3 (#) g1 is_differentiable_in t0 & f1 (#) g3 is_differentiable_in t0 & f1 (#) g2 is_differentiable_in t0 & f2 (#) g1 is_differentiable_in t0 ) by A1, A2, FDIFF_1:16; then VFuncdiff (((f2 (#) g3) - (f3 (#) g2)),((f3 (#) g1) - (f1 (#) g3)),((f1 (#) g2) - (f2 (#) g1)),t0) = (VFuncdiff ((f2 (#) g3),(f3 (#) g1),(f1 (#) g2),t0)) - (VFuncdiff ((f3 (#) g2),(f1 (#) g3),(f2 (#) g1),t0)) by Th98 .= (|[((g3 . t0) * (diff (f2,t0))),((g1 . t0) * (diff (f3,t0))),((g2 . t0) * (diff (f1,t0)))]| + |[((f2 . t0) * (diff (g3,t0))),((f3 . t0) * (diff (g1,t0))),((f1 . t0) * (diff (g2,t0)))]|) - (VFuncdiff ((f3 (#) g2),(f1 (#) g3),(f2 (#) g1),t0)) by A1, A2, Th100 .= (|[((g3 . t0) * (diff (f2,t0))),((g1 . t0) * (diff (f3,t0))),((g2 . t0) * (diff (f1,t0)))]| + |[((f2 . t0) * (diff (g3,t0))),((f3 . t0) * (diff (g1,t0))),((f1 . t0) * (diff (g2,t0)))]|) - (|[((g2 . t0) * (diff (f3,t0))),((g3 . t0) * (diff (f1,t0))),((g1 . t0) * (diff (f2,t0)))]| + |[((f3 . t0) * (diff (g2,t0))),((f1 . t0) * (diff (g3,t0))),((f2 . t0) * (diff (g1,t0)))]|) by A1, A2, Th100 .= (|[((g3 . t0) * (diff (f2,t0))),((g1 . t0) * (diff (f3,t0))),((g2 . t0) * (diff (f1,t0)))]| - |[((g2 . t0) * (diff (f3,t0))),((g3 . t0) * (diff (f1,t0))),((g1 . t0) * (diff (f2,t0)))]|) + (|[((f2 . t0) * (diff (g3,t0))),((f3 . t0) * (diff (g1,t0))),((f1 . t0) * (diff (g2,t0)))]| - |[((f3 . t0) * (diff (g2,t0))),((f1 . t0) * (diff (g3,t0))),((f2 . t0) * (diff (g1,t0)))]|) by Lm10 .= |[(((g3 . t0) * (diff (f2,t0))) - ((g2 . t0) * (diff (f3,t0)))),(((g1 . t0) * (diff (f3,t0))) - ((g3 . t0) * (diff (f1,t0)))),(((g2 . t0) * (diff (f1,t0))) - ((g1 . t0) * (diff (f2,t0))))]| + (|[((f2 . t0) * (diff (g3,t0))),((f3 . t0) * (diff (g1,t0))),((f1 . t0) * (diff (g2,t0)))]| - |[((f3 . t0) * (diff (g2,t0))),((f1 . t0) * (diff (g3,t0))),((f2 . t0) * (diff (g1,t0)))]|) by Lm11 .= |[(((g3 . t0) * (diff (f2,t0))) - ((g2 . t0) * (diff (f3,t0)))),(((g1 . t0) * (diff (f3,t0))) - ((g3 . t0) * (diff (f1,t0)))),(((g2 . t0) * (diff (f1,t0))) - ((g1 . t0) * (diff (f2,t0))))]| + |[(((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0)))),(((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0)))),(((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))]| by Lm11 ; hence VFuncdiff (((f2 (#) g3) - (f3 (#) g2)),((f3 (#) g1) - (f1 (#) g3)),((f1 (#) g2) - (f2 (#) g1)),t0) = |[(((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0)))),(((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0)))),(((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))]| + |[(((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0))),(((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))),(((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))]| ; ::_thesis: verum end; theorem :: EUCLID_8:112 for f1, f2, f3, g1, g2, g3, h1, h2, h3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0 holds VFuncdiff ((h1 (#) ((f2 (#) g3) - (f3 (#) g2))),(h2 (#) ((f3 (#) g1) - (f1 (#) g3))),(h3 (#) ((f1 (#) g2) - (f2 (#) g1))),t0) = (|[((diff (h1,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))),((diff (h2,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))),((diff (h3,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))]| + |[((h1 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))),((h2 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))),((h3 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))]|) + |[((h1 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))),((h2 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))),((h3 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0)))))]| proof let f1, f2, f3, g1, g2, g3, h1, h2, h3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0 holds VFuncdiff ((h1 (#) ((f2 (#) g3) - (f3 (#) g2))),(h2 (#) ((f3 (#) g1) - (f1 (#) g3))),(h3 (#) ((f1 (#) g2) - (f2 (#) g1))),t0) = (|[((diff (h1,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))),((diff (h2,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))),((diff (h3,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))]| + |[((h1 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))),((h2 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))),((h3 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))]|) + |[((h1 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))),((h2 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))),((h3 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0)))))]| let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0 implies VFuncdiff ((h1 (#) ((f2 (#) g3) - (f3 (#) g2))),(h2 (#) ((f3 (#) g1) - (f1 (#) g3))),(h3 (#) ((f1 (#) g2) - (f2 (#) g1))),t0) = (|[((diff (h1,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))),((diff (h2,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))),((diff (h3,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))]| + |[((h1 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))),((h2 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))),((h3 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))]|) + |[((h1 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))),((h2 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))),((h3 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0)))))]| ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 ) and A3: ( h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0 ) ; ::_thesis: VFuncdiff ((h1 (#) ((f2 (#) g3) - (f3 (#) g2))),(h2 (#) ((f3 (#) g1) - (f1 (#) g3))),(h3 (#) ((f1 (#) g2) - (f2 (#) g1))),t0) = (|[((diff (h1,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))),((diff (h2,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))),((diff (h3,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))]| + |[((h1 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))),((h2 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))),((h3 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))]|) + |[((h1 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))),((h2 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))),((h3 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0)))))]| A4: ( f2 (#) g3 is_differentiable_in t0 & f3 (#) g2 is_differentiable_in t0 & f3 (#) g1 is_differentiable_in t0 & f1 (#) g3 is_differentiable_in t0 & f1 (#) g2 is_differentiable_in t0 & f2 (#) g1 is_differentiable_in t0 ) by A1, A2, FDIFF_1:16; then A5: ( (f2 (#) g3) - (f3 (#) g2) is_differentiable_in t0 & (f3 (#) g1) - (f1 (#) g3) is_differentiable_in t0 & (f1 (#) g2) - (f2 (#) g1) is_differentiable_in t0 ) by FDIFF_1:14; then A6: (f2 (#) g3) - (f3 (#) g2) is_left_differentiable_in t0 by FDIFF_3:22; A7: t0 in dom ((f2 (#) g3) - (f3 (#) g2)) proof consider r being Element of REAL such that A8: ( 0 < r & [.(t0 - r),t0.] c= dom ((f2 (#) g3) - (f3 (#) g2)) ) by A6, FDIFF_3:def_4; t0 - r <= t0 by A8, XREAL_1:44; then t0 in [.(t0 - r),t0.] ; hence t0 in dom ((f2 (#) g3) - (f3 (#) g2)) by A8; ::_thesis: verum end; A9: (f3 (#) g1) - (f1 (#) g3) is_left_differentiable_in t0 by A5, FDIFF_3:22; A10: t0 in dom ((f3 (#) g1) - (f1 (#) g3)) proof consider r1 being Element of REAL such that A11: ( 0 < r1 & [.(t0 - r1),t0.] c= dom ((f3 (#) g1) - (f1 (#) g3)) ) by A9, FDIFF_3:def_4; t0 - r1 <= t0 by A11, XREAL_1:44; then t0 in [.(t0 - r1),t0.] ; hence t0 in dom ((f3 (#) g1) - (f1 (#) g3)) by A11; ::_thesis: verum end; A12: (f1 (#) g2) - (f2 (#) g1) is_left_differentiable_in t0 by A5, FDIFF_3:22; A13: t0 in dom ((f1 (#) g2) - (f2 (#) g1)) proof consider r2 being Element of REAL such that A14: ( 0 < r2 & [.(t0 - r2),t0.] c= dom ((f1 (#) g2) - (f2 (#) g1)) ) by A12, FDIFF_3:def_4; t0 - r2 <= t0 by A14, XREAL_1:44; then t0 in [.(t0 - r2),t0.] ; hence t0 in dom ((f1 (#) g2) - (f2 (#) g1)) by A14; ::_thesis: verum end; VFuncdiff ((h1 (#) ((f2 (#) g3) - (f3 (#) g2))),(h2 (#) ((f3 (#) g1) - (f1 (#) g3))),(h3 (#) ((f1 (#) g2) - (f2 (#) g1))),t0) = |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h1,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h2 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h3 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0)))]| by A3, A5, Th100 .= |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h1,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0)))),((h2 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h3 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0)))]| by A4, FDIFF_1:14 .= |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h1,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0)))),((h2 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0)))),((h3 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0)))]| by A4, FDIFF_1:14 .= |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h1,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0)))),((h2 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0)))),((h3 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0))))]| by A4, FDIFF_1:14 .= |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h1,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (diff ((f3 (#) g2),t0)))),((h2 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0)))),((h3 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0))))]| by A1, A2, FDIFF_1:16 .= |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h1,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h2 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0)))),((h3 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0))))]| by A1, A2, FDIFF_1:16 .= |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h1,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h2 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (diff ((f1 (#) g3),t0)))),((h3 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0))))]| by A1, A2, FDIFF_1:16 .= |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h1,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h2 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h3 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0))))]| by A1, A2, FDIFF_1:16 .= |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h1,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h2 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h3 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (diff ((f2 (#) g1),t0))))]| by A1, A2, FDIFF_1:16 .= |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h1,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h2 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h3 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0))))))]| by A1, A2, FDIFF_1:16 .= |[((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h1,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h2 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h3 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0))))))]| by A7, VALUED_1:13 .= |[((((f2 . t0) * (g3 . t0)) - ((f3 (#) g2) . t0)) * (diff (h1,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h2 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h3 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0))))))]| by VALUED_1:5 .= |[((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h1,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h2 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h3 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0))))))]| by VALUED_1:5 .= |[((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h1,t0))),((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h2 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h3 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0))))))]| by A10, VALUED_1:13 .= |[((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h1,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 (#) g3) . t0)) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h2 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h3 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0))))))]| by VALUED_1:5 .= |[((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h1,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h2,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h2 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h3 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0))))))]| by VALUED_1:5 .= |[((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h1,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h2,t0))),((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h2 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h3 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0))))))]| by A13, VALUED_1:13 .= |[((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h1,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h2,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 (#) g1) . t0)) * (diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h2 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h3 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0))))))]| by VALUED_1:5 .= |[((diff (h1,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))),((diff (h2,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))),((diff (h3,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))]| + |[(((((h1 . t0) * (g3 . t0)) * (diff (f2,t0))) - (((h1 . t0) * (g2 . t0)) * (diff (f3,t0)))) + ((((h1 . t0) * (f2 . t0)) * (diff (g3,t0))) - (((h1 . t0) * (f3 . t0)) * (diff (g2,t0))))),(((((h2 . t0) * (g1 . t0)) * (diff (f3,t0))) - (((h2 . t0) * (g3 . t0)) * (diff (f1,t0)))) + ((((h2 . t0) * (f3 . t0)) * (diff (g1,t0))) - (((h2 . t0) * (f1 . t0)) * (diff (g3,t0))))),(((((h3 . t0) * (g2 . t0)) * (diff (f1,t0))) - (((h3 . t0) * (g1 . t0)) * (diff (f2,t0)))) + ((((h3 . t0) * (f1 . t0)) * (diff (g2,t0))) - (((h3 . t0) * (f2 . t0)) * (diff (g1,t0)))))]| by VALUED_1:5 .= |[((diff (h1,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))),((diff (h2,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))),((diff (h3,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))]| + (|[((((h1 . t0) * (g3 . t0)) * (diff (f2,t0))) - (((h1 . t0) * (g2 . t0)) * (diff (f3,t0)))),((((h2 . t0) * (g1 . t0)) * (diff (f3,t0))) - (((h2 . t0) * (g3 . t0)) * (diff (f1,t0)))),((((h3 . t0) * (g2 . t0)) * (diff (f1,t0))) - (((h3 . t0) * (g1 . t0)) * (diff (f2,t0))))]| + |[((((h1 . t0) * (f2 . t0)) * (diff (g3,t0))) - (((h1 . t0) * (f3 . t0)) * (diff (g2,t0)))),((((h2 . t0) * (f3 . t0)) * (diff (g1,t0))) - (((h2 . t0) * (f1 . t0)) * (diff (g3,t0)))),((((h3 . t0) * (f1 . t0)) * (diff (g2,t0))) - (((h3 . t0) * (f2 . t0)) * (diff (g1,t0))))]|) by Lm8 .= (|[((diff (h1,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))),((diff (h2,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))),((diff (h3,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))]| + |[((h1 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))),((h2 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))),((h3 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))]|) + |[((h1 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))),((h2 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))),((h3 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0)))))]| by RVSUM_1:15 ; hence VFuncdiff ((h1 (#) ((f2 (#) g3) - (f3 (#) g2))),(h2 (#) ((f3 (#) g1) - (f1 (#) g3))),(h3 (#) ((f1 (#) g2) - (f2 (#) g1))),t0) = (|[((diff (h1,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))),((diff (h2,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))),((diff (h3,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))]| + |[((h1 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))),((h2 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))),((h3 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))]|) + |[((h1 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))),((h2 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))),((h3 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0)))))]| ; ::_thesis: verum end; theorem :: EUCLID_8:113 for f1, f2, f3, g1, g2, g3, h1, h2, h3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0 holds VFuncdiff ((((h2 (#) f2) (#) g3) - ((h3 (#) f3) (#) g2)),(((h3 (#) f3) (#) g1) - ((h1 (#) f1) (#) g3)),(((h1 (#) f1) (#) g2) - ((h2 (#) f2) (#) g1)),t0) = (|[((((h2 . t0) * (f2 . t0)) * (diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * (diff (g2,t0)))),((((h3 . t0) * (f3 . t0)) * (diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * (diff (g3,t0)))),((((h1 . t0) * (f1 . t0)) * (diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * (diff (g1,t0))))]| + |[((((h2 . t0) * (diff (f2,t0))) * (g3 . t0)) - (((h3 . t0) * (diff (f3,t0))) * (g2 . t0))),((((h3 . t0) * (diff (f3,t0))) * (g1 . t0)) - (((h1 . t0) * (diff (f1,t0))) * (g3 . t0))),((((h1 . t0) * (diff (f1,t0))) * (g2 . t0)) - (((h2 . t0) * (diff (f2,t0))) * (g1 . t0)))]|) + |[((((diff (h2,t0)) * (f2 . t0)) * (g3 . t0)) - (((diff (h3,t0)) * (f3 . t0)) * (g2 . t0))),((((diff (h3,t0)) * (f3 . t0)) * (g1 . t0)) - (((diff (h1,t0)) * (f1 . t0)) * (g3 . t0))),((((diff (h1,t0)) * (f1 . t0)) * (g2 . t0)) - (((diff (h2,t0)) * (f2 . t0)) * (g1 . t0)))]| proof let f1, f2, f3, g1, g2, g3, h1, h2, h3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0 holds VFuncdiff ((((h2 (#) f2) (#) g3) - ((h3 (#) f3) (#) g2)),(((h3 (#) f3) (#) g1) - ((h1 (#) f1) (#) g3)),(((h1 (#) f1) (#) g2) - ((h2 (#) f2) (#) g1)),t0) = (|[((((h2 . t0) * (f2 . t0)) * (diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * (diff (g2,t0)))),((((h3 . t0) * (f3 . t0)) * (diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * (diff (g3,t0)))),((((h1 . t0) * (f1 . t0)) * (diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * (diff (g1,t0))))]| + |[((((h2 . t0) * (diff (f2,t0))) * (g3 . t0)) - (((h3 . t0) * (diff (f3,t0))) * (g2 . t0))),((((h3 . t0) * (diff (f3,t0))) * (g1 . t0)) - (((h1 . t0) * (diff (f1,t0))) * (g3 . t0))),((((h1 . t0) * (diff (f1,t0))) * (g2 . t0)) - (((h2 . t0) * (diff (f2,t0))) * (g1 . t0)))]|) + |[((((diff (h2,t0)) * (f2 . t0)) * (g3 . t0)) - (((diff (h3,t0)) * (f3 . t0)) * (g2 . t0))),((((diff (h3,t0)) * (f3 . t0)) * (g1 . t0)) - (((diff (h1,t0)) * (f1 . t0)) * (g3 . t0))),((((diff (h1,t0)) * (f1 . t0)) * (g2 . t0)) - (((diff (h2,t0)) * (f2 . t0)) * (g1 . t0)))]| let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0 implies VFuncdiff ((((h2 (#) f2) (#) g3) - ((h3 (#) f3) (#) g2)),(((h3 (#) f3) (#) g1) - ((h1 (#) f1) (#) g3)),(((h1 (#) f1) (#) g2) - ((h2 (#) f2) (#) g1)),t0) = (|[((((h2 . t0) * (f2 . t0)) * (diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * (diff (g2,t0)))),((((h3 . t0) * (f3 . t0)) * (diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * (diff (g3,t0)))),((((h1 . t0) * (f1 . t0)) * (diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * (diff (g1,t0))))]| + |[((((h2 . t0) * (diff (f2,t0))) * (g3 . t0)) - (((h3 . t0) * (diff (f3,t0))) * (g2 . t0))),((((h3 . t0) * (diff (f3,t0))) * (g1 . t0)) - (((h1 . t0) * (diff (f1,t0))) * (g3 . t0))),((((h1 . t0) * (diff (f1,t0))) * (g2 . t0)) - (((h2 . t0) * (diff (f2,t0))) * (g1 . t0)))]|) + |[((((diff (h2,t0)) * (f2 . t0)) * (g3 . t0)) - (((diff (h3,t0)) * (f3 . t0)) * (g2 . t0))),((((diff (h3,t0)) * (f3 . t0)) * (g1 . t0)) - (((diff (h1,t0)) * (f1 . t0)) * (g3 . t0))),((((diff (h1,t0)) * (f1 . t0)) * (g2 . t0)) - (((diff (h2,t0)) * (f2 . t0)) * (g1 . t0)))]| ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 ) and A3: ( h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0 ) ; ::_thesis: VFuncdiff ((((h2 (#) f2) (#) g3) - ((h3 (#) f3) (#) g2)),(((h3 (#) f3) (#) g1) - ((h1 (#) f1) (#) g3)),(((h1 (#) f1) (#) g2) - ((h2 (#) f2) (#) g1)),t0) = (|[((((h2 . t0) * (f2 . t0)) * (diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * (diff (g2,t0)))),((((h3 . t0) * (f3 . t0)) * (diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * (diff (g3,t0)))),((((h1 . t0) * (f1 . t0)) * (diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * (diff (g1,t0))))]| + |[((((h2 . t0) * (diff (f2,t0))) * (g3 . t0)) - (((h3 . t0) * (diff (f3,t0))) * (g2 . t0))),((((h3 . t0) * (diff (f3,t0))) * (g1 . t0)) - (((h1 . t0) * (diff (f1,t0))) * (g3 . t0))),((((h1 . t0) * (diff (f1,t0))) * (g2 . t0)) - (((h2 . t0) * (diff (f2,t0))) * (g1 . t0)))]|) + |[((((diff (h2,t0)) * (f2 . t0)) * (g3 . t0)) - (((diff (h3,t0)) * (f3 . t0)) * (g2 . t0))),((((diff (h3,t0)) * (f3 . t0)) * (g1 . t0)) - (((diff (h1,t0)) * (f1 . t0)) * (g3 . t0))),((((diff (h1,t0)) * (f1 . t0)) * (g2 . t0)) - (((diff (h2,t0)) * (f2 . t0)) * (g1 . t0)))]| A4: ( h3 (#) f3 is_differentiable_in t0 & h1 (#) f1 is_differentiable_in t0 & h2 (#) f2 is_differentiable_in t0 ) by A1, A3, FDIFF_1:16; then A5: ( (h3 (#) f3) (#) g1 is_differentiable_in t0 & (h3 (#) f3) (#) g2 is_differentiable_in t0 & (h1 (#) f1) (#) g2 is_differentiable_in t0 & (h1 (#) f1) (#) g3 is_differentiable_in t0 & (h2 (#) f2) (#) g3 is_differentiable_in t0 & (h2 (#) f2) (#) g1 is_differentiable_in t0 ) by A2, FDIFF_1:16; VFuncdiff ((((h2 (#) f2) (#) g3) - ((h3 (#) f3) (#) g2)),(((h3 (#) f3) (#) g1) - ((h1 (#) f1) (#) g3)),(((h1 (#) f1) (#) g2) - ((h2 (#) f2) (#) g1)),t0) = (VFuncdiff (((h2 (#) f2) (#) g3),((h3 (#) f3) (#) g1),((h1 (#) f1) (#) g2),t0)) - (VFuncdiff (((h3 (#) f3) (#) g2),((h1 (#) f1) (#) g3),((h2 (#) f2) (#) g1),t0)) by A5, Th98 .= (|[((g3 . t0) * (diff ((h2 (#) f2),t0))),((g1 . t0) * (diff ((h3 (#) f3),t0))),((g2 . t0) * (diff ((h1 (#) f1),t0)))]| + |[(((h2 (#) f2) . t0) * (diff (g3,t0))),(((h3 (#) f3) . t0) * (diff (g1,t0))),(((h1 (#) f1) . t0) * (diff (g2,t0)))]|) - (VFuncdiff (((h3 (#) f3) (#) g2),((h1 (#) f1) (#) g3),((h2 (#) f2) (#) g1),t0)) by A2, A4, Th100 .= (|[((g3 . t0) * (diff ((h2 (#) f2),t0))),((g1 . t0) * (diff ((h3 (#) f3),t0))),((g2 . t0) * (diff ((h1 (#) f1),t0)))]| + |[(((h2 (#) f2) . t0) * (diff (g3,t0))),(((h3 (#) f3) . t0) * (diff (g1,t0))),(((h1 (#) f1) . t0) * (diff (g2,t0)))]|) - (|[((g2 . t0) * (diff ((h3 (#) f3),t0))),((g3 . t0) * (diff ((h1 (#) f1),t0))),((g1 . t0) * (diff ((h2 (#) f2),t0)))]| + |[(((h3 (#) f3) . t0) * (diff (g2,t0))),(((h1 (#) f1) . t0) * (diff (g3,t0))),(((h2 (#) f2) . t0) * (diff (g1,t0)))]|) by A2, A4, Th100 .= (|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (diff ((h3 (#) f3),t0))),((g2 . t0) * (diff ((h1 (#) f1),t0)))]| + |[(((h2 (#) f2) . t0) * (diff (g3,t0))),(((h3 (#) f3) . t0) * (diff (g1,t0))),(((h1 (#) f1) . t0) * (diff (g2,t0)))]|) - (|[((g2 . t0) * (diff ((h3 (#) f3),t0))),((g3 . t0) * (diff ((h1 (#) f1),t0))),((g1 . t0) * (diff ((h2 (#) f2),t0)))]| + |[(((h3 (#) f3) . t0) * (diff (g2,t0))),(((h1 (#) f1) . t0) * (diff (g3,t0))),(((h2 (#) f2) . t0) * (diff (g1,t0)))]|) by A1, A3, FDIFF_1:16 .= (|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (diff ((h1 (#) f1),t0)))]| + |[(((h2 (#) f2) . t0) * (diff (g3,t0))),(((h3 (#) f3) . t0) * (diff (g1,t0))),(((h1 (#) f1) . t0) * (diff (g2,t0)))]|) - (|[((g2 . t0) * (diff ((h3 (#) f3),t0))),((g3 . t0) * (diff ((h1 (#) f1),t0))),((g1 . t0) * (diff ((h2 (#) f2),t0)))]| + |[(((h3 (#) f3) . t0) * (diff (g2,t0))),(((h1 (#) f1) . t0) * (diff (g3,t0))),(((h2 (#) f2) . t0) * (diff (g1,t0)))]|) by A1, A3, FDIFF_1:16 .= (|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))]| + |[(((h2 (#) f2) . t0) * (diff (g3,t0))),(((h3 (#) f3) . t0) * (diff (g1,t0))),(((h1 (#) f1) . t0) * (diff (g2,t0)))]|) - (|[((g2 . t0) * (diff ((h3 (#) f3),t0))),((g3 . t0) * (diff ((h1 (#) f1),t0))),((g1 . t0) * (diff ((h2 (#) f2),t0)))]| + |[(((h3 (#) f3) . t0) * (diff (g2,t0))),(((h1 (#) f1) . t0) * (diff (g3,t0))),(((h2 (#) f2) . t0) * (diff (g1,t0)))]|) by A1, A3, FDIFF_1:16 .= (|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * (diff (g3,t0))),(((h3 (#) f3) . t0) * (diff (g1,t0))),(((h1 (#) f1) . t0) * (diff (g2,t0)))]|) - (|[((g2 . t0) * (diff ((h3 (#) f3),t0))),((g3 . t0) * (diff ((h1 (#) f1),t0))),((g1 . t0) * (diff ((h2 (#) f2),t0)))]| + |[(((h3 (#) f3) . t0) * (diff (g2,t0))),(((h1 (#) f1) . t0) * (diff (g3,t0))),(((h2 (#) f2) . t0) * (diff (g1,t0)))]|) by VALUED_1:5 .= (|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * (diff (g3,t0))),(((h3 . t0) * (f3 . t0)) * (diff (g1,t0))),(((h1 (#) f1) . t0) * (diff (g2,t0)))]|) - (|[((g2 . t0) * (diff ((h3 (#) f3),t0))),((g3 . t0) * (diff ((h1 (#) f1),t0))),((g1 . t0) * (diff ((h2 (#) f2),t0)))]| + |[(((h3 (#) f3) . t0) * (diff (g2,t0))),(((h1 (#) f1) . t0) * (diff (g3,t0))),(((h2 (#) f2) . t0) * (diff (g1,t0)))]|) by VALUED_1:5 .= (|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * (diff (g3,t0))),(((h3 . t0) * (f3 . t0)) * (diff (g1,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g2,t0)))]|) - (|[((g2 . t0) * (diff ((h3 (#) f3),t0))),((g3 . t0) * (diff ((h1 (#) f1),t0))),((g1 . t0) * (diff ((h2 (#) f2),t0)))]| + |[(((h3 (#) f3) . t0) * (diff (g2,t0))),(((h1 (#) f1) . t0) * (diff (g3,t0))),(((h2 (#) f2) . t0) * (diff (g1,t0)))]|) by VALUED_1:5 .= (|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * (diff (g3,t0))),(((h3 . t0) * (f3 . t0)) * (diff (g1,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g2,t0)))]|) - (|[((g2 . t0) * (diff ((h3 (#) f3),t0))),((g3 . t0) * (diff ((h1 (#) f1),t0))),((g1 . t0) * (diff ((h2 (#) f2),t0)))]| + |[(((h3 . t0) * (f3 . t0)) * (diff (g2,t0))),(((h1 (#) f1) . t0) * (diff (g3,t0))),(((h2 (#) f2) . t0) * (diff (g1,t0)))]|) by VALUED_1:5 .= (|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * (diff (g3,t0))),(((h3 . t0) * (f3 . t0)) * (diff (g1,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g2,t0)))]|) - (|[((g2 . t0) * (diff ((h3 (#) f3),t0))),((g3 . t0) * (diff ((h1 (#) f1),t0))),((g1 . t0) * (diff ((h2 (#) f2),t0)))]| + |[(((h3 . t0) * (f3 . t0)) * (diff (g2,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g3,t0))),(((h2 (#) f2) . t0) * (diff (g1,t0)))]|) by VALUED_1:5 .= (|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * (diff (g3,t0))),(((h3 . t0) * (f3 . t0)) * (diff (g1,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g2,t0)))]|) - (|[((g2 . t0) * (diff ((h3 (#) f3),t0))),((g3 . t0) * (diff ((h1 (#) f1),t0))),((g1 . t0) * (diff ((h2 (#) f2),t0)))]| + |[(((h3 . t0) * (f3 . t0)) * (diff (g2,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g3,t0))),(((h2 . t0) * (f2 . t0)) * (diff (g1,t0)))]|) by VALUED_1:5 .= (|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * (diff (g3,t0))),(((h3 . t0) * (f3 . t0)) * (diff (g1,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g2,t0)))]|) - (|[((g2 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g3 . t0) * (diff ((h1 (#) f1),t0))),((g1 . t0) * (diff ((h2 (#) f2),t0)))]| + |[(((h3 . t0) * (f3 . t0)) * (diff (g2,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g3,t0))),(((h2 . t0) * (f2 . t0)) * (diff (g1,t0)))]|) by A1, A3, FDIFF_1:16 .= (|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * (diff (g3,t0))),(((h3 . t0) * (f3 . t0)) * (diff (g1,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g2,t0)))]|) - (|[((g2 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g3 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0))))),((g1 . t0) * (diff ((h2 (#) f2),t0)))]| + |[(((h3 . t0) * (f3 . t0)) * (diff (g2,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g3,t0))),(((h2 . t0) * (f2 . t0)) * (diff (g1,t0)))]|) by A1, A3, FDIFF_1:16 .= (|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * (diff (g3,t0))),(((h3 . t0) * (f3 . t0)) * (diff (g1,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g2,t0)))]|) - (|[((g2 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g3 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0))))),((g1 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0)))))]| + |[(((h3 . t0) * (f3 . t0)) * (diff (g2,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g3,t0))),(((h2 . t0) * (f2 . t0)) * (diff (g1,t0)))]|) by A1, A3, FDIFF_1:16 .= ((|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * (diff (g3,t0))),(((h3 . t0) * (f3 . t0)) * (diff (g1,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g2,t0)))]|) - |[((g2 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g3 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0))))),((g1 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0)))))]|) - |[(((h3 . t0) * (f3 . t0)) * (diff (g2,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g3,t0))),(((h2 . t0) * (f2 . t0)) * (diff (g1,t0)))]| by RVSUM_1:39 .= ((|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))]| - |[((g2 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g3 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0))))),((g1 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0)))))]|) + |[(((h2 . t0) * (f2 . t0)) * (diff (g3,t0))),(((h3 . t0) * (f3 . t0)) * (diff (g1,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g2,t0)))]|) + (- |[(((h3 . t0) * (f3 . t0)) * (diff (g2,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g3,t0))),(((h2 . t0) * (f2 . t0)) * (diff (g1,t0)))]|) by RVSUM_1:15 .= (|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))]| - |[((g2 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g3 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0))))),((g1 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0)))))]|) + (|[(((h2 . t0) * (f2 . t0)) * (diff (g3,t0))),(((h3 . t0) * (f3 . t0)) * (diff (g1,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g2,t0)))]| - |[(((h3 . t0) * (f3 . t0)) * (diff (g2,t0))),(((h1 . t0) * (f1 . t0)) * (diff (g3,t0))),(((h2 . t0) * (f2 . t0)) * (diff (g1,t0)))]|) by RVSUM_1:15 .= (|[((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))),((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))]| - |[((g2 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))),((g3 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0))))),((g1 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0)))))]|) + |[((((h2 . t0) * (f2 . t0)) * (diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * (diff (g2,t0)))),((((h3 . t0) * (f3 . t0)) * (diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * (diff (g3,t0)))),((((h1 . t0) * (f1 . t0)) * (diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * (diff (g1,t0))))]| by Lm11 .= |[((((h2 . t0) * (f2 . t0)) * (diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * (diff (g2,t0)))),((((h3 . t0) * (f3 . t0)) * (diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * (diff (g3,t0)))),((((h1 . t0) * (f1 . t0)) * (diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * (diff (g1,t0))))]| + |[(((g3 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))) - ((g2 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0)))))),(((g1 . t0) * (((f3 . t0) * (diff (h3,t0))) + ((h3 . t0) * (diff (f3,t0))))) - ((g3 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0)))))),(((g2 . t0) * (((f1 . t0) * (diff (h1,t0))) + ((h1 . t0) * (diff (f1,t0))))) - ((g1 . t0) * (((f2 . t0) * (diff (h2,t0))) + ((h2 . t0) * (diff (f2,t0))))))]| by Lm11 .= |[((((h2 . t0) * (f2 . t0)) * (diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * (diff (g2,t0)))),((((h3 . t0) * (f3 . t0)) * (diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * (diff (g3,t0)))),((((h1 . t0) * (f1 . t0)) * (diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * (diff (g1,t0))))]| + |[(((((g3 . t0) * (h2 . t0)) * (diff (f2,t0))) - (((g2 . t0) * (h3 . t0)) * (diff (f3,t0)))) + ((((g3 . t0) * (f2 . t0)) * (diff (h2,t0))) - (((g2 . t0) * (f3 . t0)) * (diff (h3,t0))))),(((((g1 . t0) * (h3 . t0)) * (diff (f3,t0))) - (((g3 . t0) * (h1 . t0)) * (diff (f1,t0)))) + ((((g1 . t0) * (f3 . t0)) * (diff (h3,t0))) - (((g3 . t0) * (f1 . t0)) * (diff (h1,t0))))),(((((g2 . t0) * (h1 . t0)) * (diff (f1,t0))) - (((g1 . t0) * (h2 . t0)) * (diff (f2,t0)))) + ((((g2 . t0) * (f1 . t0)) * (diff (h1,t0))) - (((g1 . t0) * (f2 . t0)) * (diff (h2,t0)))))]| .= |[((((h2 . t0) * (f2 . t0)) * (diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * (diff (g2,t0)))),((((h3 . t0) * (f3 . t0)) * (diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * (diff (g3,t0)))),((((h1 . t0) * (f1 . t0)) * (diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * (diff (g1,t0))))]| + (|[((((g3 . t0) * (h2 . t0)) * (diff (f2,t0))) - (((g2 . t0) * (h3 . t0)) * (diff (f3,t0)))),((((g1 . t0) * (h3 . t0)) * (diff (f3,t0))) - (((g3 . t0) * (h1 . t0)) * (diff (f1,t0)))),((((g2 . t0) * (h1 . t0)) * (diff (f1,t0))) - (((g1 . t0) * (h2 . t0)) * (diff (f2,t0))))]| + |[((((g3 . t0) * (f2 . t0)) * (diff (h2,t0))) - (((g2 . t0) * (f3 . t0)) * (diff (h3,t0)))),((((g1 . t0) * (f3 . t0)) * (diff (h3,t0))) - (((g3 . t0) * (f1 . t0)) * (diff (h1,t0)))),((((g2 . t0) * (f1 . t0)) * (diff (h1,t0))) - (((g1 . t0) * (f2 . t0)) * (diff (h2,t0))))]|) by Lm8 .= (|[((((h2 . t0) * (f2 . t0)) * (diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * (diff (g2,t0)))),((((h3 . t0) * (f3 . t0)) * (diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * (diff (g3,t0)))),((((h1 . t0) * (f1 . t0)) * (diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * (diff (g1,t0))))]| + |[((((g3 . t0) * (h2 . t0)) * (diff (f2,t0))) - (((g2 . t0) * (h3 . t0)) * (diff (f3,t0)))),((((g1 . t0) * (h3 . t0)) * (diff (f3,t0))) - (((g3 . t0) * (h1 . t0)) * (diff (f1,t0)))),((((g2 . t0) * (h1 . t0)) * (diff (f1,t0))) - (((g1 . t0) * (h2 . t0)) * (diff (f2,t0))))]|) + |[((((g3 . t0) * (f2 . t0)) * (diff (h2,t0))) - (((g2 . t0) * (f3 . t0)) * (diff (h3,t0)))),((((g1 . t0) * (f3 . t0)) * (diff (h3,t0))) - (((g3 . t0) * (f1 . t0)) * (diff (h1,t0)))),((((g2 . t0) * (f1 . t0)) * (diff (h1,t0))) - (((g1 . t0) * (f2 . t0)) * (diff (h2,t0))))]| by RVSUM_1:15 ; hence VFuncdiff ((((h2 (#) f2) (#) g3) - ((h3 (#) f3) (#) g2)),(((h3 (#) f3) (#) g1) - ((h1 (#) f1) (#) g3)),(((h1 (#) f1) (#) g2) - ((h2 (#) f2) (#) g1)),t0) = (|[((((h2 . t0) * (f2 . t0)) * (diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * (diff (g2,t0)))),((((h3 . t0) * (f3 . t0)) * (diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * (diff (g3,t0)))),((((h1 . t0) * (f1 . t0)) * (diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * (diff (g1,t0))))]| + |[((((h2 . t0) * (diff (f2,t0))) * (g3 . t0)) - (((h3 . t0) * (diff (f3,t0))) * (g2 . t0))),((((h3 . t0) * (diff (f3,t0))) * (g1 . t0)) - (((h1 . t0) * (diff (f1,t0))) * (g3 . t0))),((((h1 . t0) * (diff (f1,t0))) * (g2 . t0)) - (((h2 . t0) * (diff (f2,t0))) * (g1 . t0)))]|) + |[((((diff (h2,t0)) * (f2 . t0)) * (g3 . t0)) - (((diff (h3,t0)) * (f3 . t0)) * (g2 . t0))),((((diff (h3,t0)) * (f3 . t0)) * (g1 . t0)) - (((diff (h1,t0)) * (f1 . t0)) * (g3 . t0))),((((diff (h1,t0)) * (f1 . t0)) * (g2 . t0)) - (((diff (h2,t0)) * (f2 . t0)) * (g1 . t0)))]| ; ::_thesis: verum end; theorem :: EUCLID_8:114 for f1, f2, f3, g1, g2, g3, h1, h2, h3 being PartFunc of REAL,REAL for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0 holds VFuncdiff (((h2 (#) ((f1 (#) g2) - (f2 (#) g1))) - (h3 (#) ((f3 (#) g1) - (f1 (#) g3)))),((h3 (#) ((f2 (#) g3) - (f3 (#) g2))) - (h1 (#) ((f1 (#) g2) - (f2 (#) g1)))),((h1 (#) ((f3 (#) g1) - (f1 (#) g3))) - (h2 (#) ((f2 (#) g3) - (f3 (#) g2)))),t0) = (|[(((h2 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) - ((h3 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0)))))),(((h3 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))) - ((h1 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0)))))),(((h1 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) - ((h2 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))))]| + |[(((h2 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))) - ((h3 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))))),(((h3 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))) - ((h1 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h1 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))) - ((h2 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))))]|) + |[(((diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - ((diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))),(((diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - ((diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))),(((diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - ((diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]| proof let f1, f2, f3, g1, g2, g3, h1, h2, h3 be PartFunc of REAL,REAL; ::_thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0 holds VFuncdiff (((h2 (#) ((f1 (#) g2) - (f2 (#) g1))) - (h3 (#) ((f3 (#) g1) - (f1 (#) g3)))),((h3 (#) ((f2 (#) g3) - (f3 (#) g2))) - (h1 (#) ((f1 (#) g2) - (f2 (#) g1)))),((h1 (#) ((f3 (#) g1) - (f1 (#) g3))) - (h2 (#) ((f2 (#) g3) - (f3 (#) g2)))),t0) = (|[(((h2 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) - ((h3 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0)))))),(((h3 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))) - ((h1 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0)))))),(((h1 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) - ((h2 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))))]| + |[(((h2 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))) - ((h3 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))))),(((h3 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))) - ((h1 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h1 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))) - ((h2 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))))]|) + |[(((diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - ((diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))),(((diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - ((diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))),(((diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - ((diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]| let t0 be Real; ::_thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0 implies VFuncdiff (((h2 (#) ((f1 (#) g2) - (f2 (#) g1))) - (h3 (#) ((f3 (#) g1) - (f1 (#) g3)))),((h3 (#) ((f2 (#) g3) - (f3 (#) g2))) - (h1 (#) ((f1 (#) g2) - (f2 (#) g1)))),((h1 (#) ((f3 (#) g1) - (f1 (#) g3))) - (h2 (#) ((f2 (#) g3) - (f3 (#) g2)))),t0) = (|[(((h2 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) - ((h3 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0)))))),(((h3 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))) - ((h1 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0)))))),(((h1 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) - ((h2 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))))]| + |[(((h2 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))) - ((h3 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))))),(((h3 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))) - ((h1 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h1 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))) - ((h2 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))))]|) + |[(((diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - ((diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))),(((diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - ((diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))),(((diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - ((diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]| ) assume that A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and A2: ( g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 ) and A3: ( h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0 ) ; ::_thesis: VFuncdiff (((h2 (#) ((f1 (#) g2) - (f2 (#) g1))) - (h3 (#) ((f3 (#) g1) - (f1 (#) g3)))),((h3 (#) ((f2 (#) g3) - (f3 (#) g2))) - (h1 (#) ((f1 (#) g2) - (f2 (#) g1)))),((h1 (#) ((f3 (#) g1) - (f1 (#) g3))) - (h2 (#) ((f2 (#) g3) - (f3 (#) g2)))),t0) = (|[(((h2 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) - ((h3 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0)))))),(((h3 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))) - ((h1 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0)))))),(((h1 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) - ((h2 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))))]| + |[(((h2 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))) - ((h3 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))))),(((h3 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))) - ((h1 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h1 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))) - ((h2 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))))]|) + |[(((diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - ((diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))),(((diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - ((diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))),(((diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - ((diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]| A4: ( f2 (#) g3 is_differentiable_in t0 & f3 (#) g2 is_differentiable_in t0 & f3 (#) g1 is_differentiable_in t0 & f1 (#) g3 is_differentiable_in t0 & f1 (#) g2 is_differentiable_in t0 & f2 (#) g1 is_differentiable_in t0 ) by A1, A2, FDIFF_1:16; then A5: ( (f2 (#) g3) - (f3 (#) g2) is_differentiable_in t0 & (f3 (#) g1) - (f1 (#) g3) is_differentiable_in t0 & (f1 (#) g2) - (f2 (#) g1) is_differentiable_in t0 ) by FDIFF_1:14; then A6: ( h3 (#) ((f2 (#) g3) - (f3 (#) g2)) is_differentiable_in t0 & h3 (#) ((f3 (#) g1) - (f1 (#) g3)) is_differentiable_in t0 & h2 (#) ((f1 (#) g2) - (f2 (#) g1)) is_differentiable_in t0 & h2 (#) ((f2 (#) g3) - (f3 (#) g2)) is_differentiable_in t0 & h1 (#) ((f3 (#) g1) - (f1 (#) g3)) is_differentiable_in t0 & h1 (#) ((f1 (#) g2) - (f2 (#) g1)) is_differentiable_in t0 ) by A3, FDIFF_1:16; A7: (f1 (#) g2) - (f2 (#) g1) is_left_differentiable_in t0 by A5, FDIFF_3:22; A8: t0 in dom ((f1 (#) g2) - (f2 (#) g1)) proof consider r being Element of REAL such that A9: ( 0 < r & [.(t0 - r),t0.] c= dom ((f1 (#) g2) - (f2 (#) g1)) ) by A7, FDIFF_3:def_4; t0 - r <= t0 by A9, XREAL_1:44; then t0 in [.(t0 - r),t0.] ; hence t0 in dom ((f1 (#) g2) - (f2 (#) g1)) by A9; ::_thesis: verum end; A10: (f2 (#) g3) - (f3 (#) g2) is_left_differentiable_in t0 by A5, FDIFF_3:22; A11: t0 in dom ((f2 (#) g3) - (f3 (#) g2)) proof consider r1 being Element of REAL such that A12: ( 0 < r1 & [.(t0 - r1),t0.] c= dom ((f2 (#) g3) - (f3 (#) g2)) ) by A10, FDIFF_3:def_4; t0 - r1 <= t0 by A12, XREAL_1:44; then t0 in [.(t0 - r1),t0.] ; hence t0 in dom ((f2 (#) g3) - (f3 (#) g2)) by A12; ::_thesis: verum end; A13: (f3 (#) g1) - (f1 (#) g3) is_left_differentiable_in t0 by A5, FDIFF_3:22; A14: t0 in dom ((f3 (#) g1) - (f1 (#) g3)) proof consider r2 being Element of REAL such that A15: ( 0 < r2 & [.(t0 - r2),t0.] c= dom ((f3 (#) g1) - (f1 (#) g3)) ) by A13, FDIFF_3:def_4; t0 - r2 <= t0 by A15, XREAL_1:44; then t0 in [.(t0 - r2),t0.] ; hence t0 in dom ((f3 (#) g1) - (f1 (#) g3)) by A15; ::_thesis: verum end; VFuncdiff (((h2 (#) ((f1 (#) g2) - (f2 (#) g1))) - (h3 (#) ((f3 (#) g1) - (f1 (#) g3)))),((h3 (#) ((f2 (#) g3) - (f3 (#) g2))) - (h1 (#) ((f1 (#) g2) - (f2 (#) g1)))),((h1 (#) ((f3 (#) g1) - (f1 (#) g3))) - (h2 (#) ((f2 (#) g3) - (f3 (#) g2)))),t0) = (VFuncdiff ((h2 (#) ((f1 (#) g2) - (f2 (#) g1))),(h3 (#) ((f2 (#) g3) - (f3 (#) g2))),(h1 (#) ((f3 (#) g1) - (f1 (#) g3))),t0)) - (VFuncdiff ((h3 (#) ((f3 (#) g1) - (f1 (#) g3))),(h1 (#) ((f1 (#) g2) - (f2 (#) g1))),(h2 (#) ((f2 (#) g3) - (f3 (#) g2))),t0)) by A6, Th98 .= (|[((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h2,t0))),((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h3,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (VFuncdiff ((h3 (#) ((f3 (#) g1) - (f1 (#) g3))),(h1 (#) ((f1 (#) g2) - (f2 (#) g1))),(h2 (#) ((f2 (#) g3) - (f3 (#) g2))),t0)) by A3, A5, Th100 .= (|[((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h2,t0))),((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h3,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h3,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h1,t0))),((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by A3, A5, Th100 .= (|[((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h2,t0))),((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h3,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h3,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h1,t0))),((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by A8, VALUED_1:13 .= (|[((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h2,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h3,t0))),((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h3,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h1,t0))),((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by A11, VALUED_1:13 .= (|[((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h2,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h3,t0))),((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 (#) g1) - (f1 (#) g3)) . t0) * (diff (h3,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h1,t0))),((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by A14, VALUED_1:13 .= (|[((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h2,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h3,t0))),((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h3,t0))),((((f1 (#) g2) - (f2 (#) g1)) . t0) * (diff (h1,t0))),((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by A14, VALUED_1:13 .= (|[((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h2,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h3,t0))),((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h3,t0))),((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h1,t0))),((((f2 (#) g3) - (f3 (#) g2)) . t0) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by A8, VALUED_1:13 .= (|[((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h2,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h3,t0))),((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h3,t0))),((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h1,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by A11, VALUED_1:13 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 (#) g1) . t0)) * (diff (h2,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h3,t0))),((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h3,t0))),((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h1,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by VALUED_1:5 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h3,t0))),((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h3,t0))),((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h1,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by VALUED_1:5 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 (#) g2) . t0)) * (diff (h3,t0))),((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h3,t0))),((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h1,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by VALUED_1:5 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h3,t0))),((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h1,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by VALUED_1:5 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 (#) g3) . t0)) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h3,t0))),((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h1,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by VALUED_1:5 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * (diff (h3,t0))),((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h1,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by VALUED_1:5 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 (#) g3) . t0)) * (diff (h3,t0))),((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h1,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by VALUED_1:5 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * (diff (h1,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by VALUED_1:5 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 (#) g1) . t0)) * (diff (h1,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by VALUED_1:5 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by VALUED_1:5 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 (#) g2) . t0)) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by VALUED_1:5 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by VALUED_1:5 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h3 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by A4, FDIFF_1:14 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h3 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0)))),((h1 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by A4, FDIFF_1:14 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h3 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0)))),((h1 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * (diff (((f3 (#) g1) - (f1 (#) g3)),t0))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by A4, FDIFF_1:14 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h3 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0)))),((h1 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0)))),((h1 . t0) * (diff (((f1 (#) g2) - (f2 (#) g1)),t0))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by A4, FDIFF_1:14 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h3 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0)))),((h1 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0)))),((h1 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h2 . t0) * (diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|) by A4, FDIFF_1:14 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h3 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0)))),((h1 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0)))),((h1 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h2 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0))))]|) by A4, FDIFF_1:14 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (diff ((f2 (#) g1),t0)))),((h3 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0)))),((h1 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0)))),((h1 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h2 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0))))]|) by A1, A2, FDIFF_1:16 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0)))))),((h3 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0)))),((h1 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0)))),((h1 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h2 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0))))]|) by A1, A2, FDIFF_1:16 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0)))))),((h3 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (diff ((f3 (#) g2),t0)))),((h1 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0)))),((h1 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h2 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0))))]|) by A1, A2, FDIFF_1:16 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0)))))),((h3 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h1 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0)))),((h1 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h2 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0))))]|) by A1, A2, FDIFF_1:16 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0)))))),((h3 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h1 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (diff ((f1 (#) g3),t0))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0)))),((h1 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h2 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0))))]|) by A1, A2, FDIFF_1:16 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0)))))),((h3 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h1 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0))))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((diff ((f3 (#) g1),t0)) - (diff ((f1 (#) g3),t0)))),((h1 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h2 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0))))]|) by A1, A2, FDIFF_1:16 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0)))))),((h3 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h1 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0))))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (diff ((f1 (#) g3),t0)))),((h1 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h2 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0))))]|) by A1, A2, FDIFF_1:16 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0)))))),((h3 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h1 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0))))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h1 . t0) * ((diff ((f1 (#) g2),t0)) - (diff ((f2 (#) g1),t0)))),((h2 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0))))]|) by A1, A2, FDIFF_1:16 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0)))))),((h3 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h1 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0))))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h1 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (diff ((f2 (#) g1),t0)))),((h2 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0))))]|) by A1, A2, FDIFF_1:16 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0)))))),((h3 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h1 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0))))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h1 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0)))))),((h2 . t0) * ((diff ((f2 (#) g3),t0)) - (diff ((f3 (#) g2),t0))))]|) by A1, A2, FDIFF_1:16 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0)))))),((h3 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h1 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0))))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h1 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0)))))),((h2 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (diff ((f3 (#) g2),t0))))]|) by A1, A2, FDIFF_1:16 .= (|[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h2,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h3,t0))),((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0)))))),((h3 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0)))))),((h1 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0))))))]|) - (|[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * (diff (h3,t0))),((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * (diff (h1,t0))),((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * (diff (h2,t0)))]| + |[((h3 . t0) * ((((g1 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g1,t0)))) - (((g3 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g3,t0)))))),((h1 . t0) * ((((g2 . t0) * (diff (f1,t0))) + ((f1 . t0) * (diff (g2,t0)))) - (((g1 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g1,t0)))))),((h2 . t0) * ((((g3 . t0) * (diff (f2,t0))) + ((f2 . t0) * (diff (g3,t0)))) - (((g2 . t0) * (diff (f3,t0))) + ((f3 . t0) * (diff (g2,t0))))))]|) by A1, A2, FDIFF_1:16 .= ((|[(((h2 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) + ((h2 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h3 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))) + ((h3 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0))))),(((h1 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) + ((h1 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))))]| + |[((diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))),((diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))),((diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))]|) - |[((diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))),((diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))),((diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))))]|) - |[(((h3 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) + ((h3 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))))),(((h1 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) + ((h1 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h2 . t0) * (((g3 . t0) * (diff (f2,t0))) - ((g2 . t0) * (diff (f3,t0))))) + ((h2 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))))]| by RVSUM_1:39 .= (|[(((h2 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) + ((h2 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h3 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))) + ((h3 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0))))),(((h1 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) + ((h1 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))))]| + (|[((diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))),((diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))),((diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))]| - |[((diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))),((diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))),((diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))))]|)) - |[(((h3 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) + ((h3 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))))),(((h1 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) + ((h1 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h2 . t0) * (((g3 . t0) * (diff (f2,t0))) - ((g2 . t0) * (diff (f3,t0))))) + ((h2 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))))]| by RVSUM_1:15 .= (|[(((h2 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) + ((h2 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h3 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))) + ((h3 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0))))),(((h1 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) + ((h1 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))))]| + |[(((diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - ((diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))),(((diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - ((diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))),(((diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - ((diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]|) - |[(((h3 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) + ((h3 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))))),(((h1 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) + ((h1 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h2 . t0) * (((g3 . t0) * (diff (f2,t0))) - ((g2 . t0) * (diff (f3,t0))))) + ((h2 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))))]| by Lm11 .= (|[(((h2 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) + ((h2 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h3 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))) + ((h3 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0))))),(((h1 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) + ((h1 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))))]| - |[(((h3 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) + ((h3 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))))),(((h1 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) + ((h1 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h2 . t0) * (((g3 . t0) * (diff (f2,t0))) - ((g2 . t0) * (diff (f3,t0))))) + ((h2 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))))]|) + |[(((diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - ((diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))),(((diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - ((diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))),(((diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - ((diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]| by RVSUM_1:15 .= |[((((h2 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) + ((h2 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))) - (((h3 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) + ((h3 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))))),((((h3 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))) + ((h3 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0))))) - (((h1 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) + ((h1 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))))),((((h1 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) + ((h1 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))))) - (((h2 . t0) * (((g3 . t0) * (diff (f2,t0))) - ((g2 . t0) * (diff (f3,t0))))) + ((h2 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0)))))))]| + |[(((diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - ((diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))),(((diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - ((diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))),(((diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - ((diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]| by Lm11 .= |[((((h2 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) - ((h3 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0)))))) + (((h2 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))) - ((h3 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))))),((((h3 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))) - ((h1 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0)))))) + (((h3 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))) - ((h1 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))))),((((h1 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) - ((h2 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0)))))) + (((h1 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))) - ((h2 . t0) * (((g3 . t0) * (diff (f2,t0))) - ((g2 . t0) * (diff (f3,t0)))))))]| + |[(((diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - ((diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))),(((diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - ((diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))),(((diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - ((diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]| .= (|[(((h2 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) - ((h3 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0)))))),(((h3 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))) - ((h1 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0)))))),(((h1 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) - ((h2 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))))]| + |[(((h2 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))) - ((h3 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))))),(((h3 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))) - ((h1 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h1 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))) - ((h2 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))))]|) + |[(((diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - ((diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))),(((diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - ((diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))),(((diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - ((diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]| by Lm8 ; hence VFuncdiff (((h2 (#) ((f1 (#) g2) - (f2 (#) g1))) - (h3 (#) ((f3 (#) g1) - (f1 (#) g3)))),((h3 (#) ((f2 (#) g3) - (f3 (#) g2))) - (h1 (#) ((f1 (#) g2) - (f2 (#) g1)))),((h1 (#) ((f3 (#) g1) - (f1 (#) g3))) - (h2 (#) ((f2 (#) g3) - (f3 (#) g2)))),t0) = (|[(((h2 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) - ((h3 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0)))))),(((h3 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))) - ((h1 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0)))))),(((h1 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) - ((h2 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))))]| + |[(((h2 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))) - ((h3 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))))),(((h3 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))) - ((h1 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h1 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))) - ((h2 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))))]|) + |[(((diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - ((diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))),(((diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - ((diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))),(((diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - ((diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]| ; ::_thesis: verum end;