let a, b, c, d be Real; :: thesis: ( a < b & c < d implies ex f being Function of I[01],(() | (Lower_Arc (rectangle (a,b,c,d)))) st
( f is being_homeomorphism & f . 0 = E-max (rectangle (a,b,c,d)) & f . 1 = W-min (rectangle (a,b,c,d)) & rng f = Lower_Arc (rectangle (a,b,c,d)) & ( for r being Real st r in [.0,(1 / 2).] holds
f . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) ) & ( for r being Real st r in [.(1 / 2),1.] holds
f . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) ) & ( for p being Point of () st p in LSeg (|[b,d]|,|[b,c]|) holds
( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f . ((((p `2) - d) / (c - d)) / 2) = p ) ) & ( for p being Point of () st p in LSeg (|[b,c]|,|[a,c]|) holds
( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) ) ) )

set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d ; :: thesis: ex f being Function of I[01],(() | (Lower_Arc (rectangle (a,b,c,d)))) st
( f is being_homeomorphism & f . 0 = E-max (rectangle (a,b,c,d)) & f . 1 = W-min (rectangle (a,b,c,d)) & rng f = Lower_Arc (rectangle (a,b,c,d)) & ( for r being Real st r in [.0,(1 / 2).] holds
f . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) ) & ( for r being Real st r in [.(1 / 2),1.] holds
f . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) ) & ( for p being Point of () st p in LSeg (|[b,d]|,|[b,c]|) holds
( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f . ((((p `2) - d) / (c - d)) / 2) = p ) ) & ( for p being Point of () st p in LSeg (|[b,c]|,|[a,c]|) holds
( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) ) )

defpred S1[ object , object ] means for r being Real st \$1 = r holds
( ( r in [.0,(1 / 2).] implies \$2 = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) ) & ( r in [.(1 / 2),1.] implies \$2 = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) ) );
A3: [.0,1.] = [.0,(1 / 2).] \/ [.(1 / 2),1.] by XXREAL_1:165;
A4: for x being object st x in [.0,1.] holds
ex y being object st S1[x,y]
proof
let x be object ; :: thesis: ( x in [.0,1.] implies ex y being object st S1[x,y] )
assume A5: x in [.0,1.] ; :: thesis: ex y being object st S1[x,y]
now :: thesis: ( ( x in [.0,(1 / 2).] & ex y being object st S1[x,y] ) or ( x in [.(1 / 2),1.] & ex y being object st S1[x,y] ) )
per cases ( x in [.0,(1 / 2).] or x in [.(1 / 2),1.] ) by ;
case A6: x in [.0,(1 / 2).] ; :: thesis: ex y being object st S1[x,y]
then reconsider r = x as Real ;
A7: r <= 1 / 2 by ;
set y0 = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|);
( r in [.(1 / 2),1.] implies ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) )
proof
assume r in [.(1 / 2),1.] ; :: thesis: ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|)
then 1 / 2 <= r by XXREAL_1:1;
then A8: r = 1 / 2 by ;
then A9: ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = (0 * |[b,d]|) + |[b,c]| by RLVECT_1:def 8
.= (0. ()) + |[b,c]| by RLVECT_1:10
.= |[b,c]| by RLVECT_1:4 ;
((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = (1 * |[b,c]|) + (0. ()) by
.= |[b,c]| + (0. ()) by RLVECT_1:def 8
.= |[b,c]| by RLVECT_1:4 ;
hence ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) by A9; :: thesis: verum
end;
then for r2 being Real st x = r2 holds
( ( r2 in [.0,(1 / 2).] implies ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = ((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|) ) & ( r2 in [.(1 / 2),1.] implies ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = ((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|) ) ) ;
hence ex y being object st S1[x,y] ; :: thesis: verum
end;
case A10: x in [.(1 / 2),1.] ; :: thesis: ex y being object st S1[x,y]
then reconsider r = x as Real ;
A11: 1 / 2 <= r by ;
set y0 = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|);
( r in [.0,(1 / 2).] implies ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) )
proof
assume r in [.0,(1 / 2).] ; :: thesis: ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|)
then r <= 1 / 2 by XXREAL_1:1;
then A12: r = 1 / 2 by ;
then A13: ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = |[b,c]| + (0 * |[a,c]|) by RLVECT_1:def 8
.= |[b,c]| + (0. ()) by RLVECT_1:10
.= |[b,c]| by RLVECT_1:4 ;
((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = (0. ()) + (1 * |[b,c]|) by
.= (0. ()) + |[b,c]| by RLVECT_1:def 8
.= |[b,c]| by RLVECT_1:4 ;
hence ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) by A13; :: thesis: verum
end;
then for r2 being Real st x = r2 holds
( ( r2 in [.0,(1 / 2).] implies ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = ((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|) ) & ( r2 in [.(1 / 2),1.] implies ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = ((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|) ) ) ;
hence ex y being object st S1[x,y] ; :: thesis: verum
end;
end;
end;
hence ex y being object st S1[x,y] ; :: thesis: verum
end;
ex f2 being Function st
( dom f2 = [.0,1.] & ( for x being object st x in [.0,1.] holds
S1[x,f2 . x] ) ) from then consider f2 being Function such that
A14: dom f2 = [.0,1.] and
A15: for x being object st x in [.0,1.] holds
S1[x,f2 . x] ;
rng f2 c= the carrier of (() | (Lower_Arc (rectangle (a,b,c,d))))
proof
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng f2 or y in the carrier of (() | (Lower_Arc (rectangle (a,b,c,d)))) )
assume y in rng f2 ; :: thesis: y in the carrier of (() | (Lower_Arc (rectangle (a,b,c,d))))
then consider x being object such that
A16: x in dom f2 and
A17: y = f2 . x by FUNCT_1:def 3;
now :: thesis: ( ( x in [.0,(1 / 2).] & y in the carrier of (() | (Lower_Arc (rectangle (a,b,c,d)))) ) or ( x in [.(1 / 2),1.] & y in the carrier of (() | (Lower_Arc (rectangle (a,b,c,d)))) ) )
per cases ( x in [.0,(1 / 2).] or x in [.(1 / 2),1.] ) by ;
case A18: x in [.0,(1 / 2).] ; :: thesis: y in the carrier of (() | (Lower_Arc (rectangle (a,b,c,d))))
then reconsider r = x as Real ;
A19: 0 <= r by ;
r <= 1 / 2 by ;
then A20: r * 2 <= (1 / 2) * 2 by XREAL_1:64;
f2 . x = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) by A14, A15, A16, A18;
then A21: y in LSeg (|[b,d]|,|[b,c]|) by ;
Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) by A1, A2, Th52;
then y in Lower_Arc (rectangle (a,b,c,d)) by ;
hence y in the carrier of (() | (Lower_Arc (rectangle (a,b,c,d)))) by PRE_TOPC:8; :: thesis: verum
end;
case A22: x in [.(1 / 2),1.] ; :: thesis: y in the carrier of (() | (Lower_Arc (rectangle (a,b,c,d))))
then reconsider r = x as Real ;
A23: 1 / 2 <= r by ;
A24: r <= 1 by ;
r * 2 >= (1 / 2) * 2 by ;
then A25: (2 * r) - 1 >= 0 by XREAL_1:48;
2 * 1 >= 2 * r by ;
then A26: (1 + 1) - 1 >= (2 * r) - 1 by XREAL_1:9;
f2 . x = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) by A14, A15, A16, A22;
then A27: y in LSeg (|[b,c]|,|[a,c]|) by ;
Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) by A1, A2, Th52;
then y in Lower_Arc (rectangle (a,b,c,d)) by ;
hence y in the carrier of (() | (Lower_Arc (rectangle (a,b,c,d)))) by PRE_TOPC:8; :: thesis: verum
end;
end;
end;
hence y in the carrier of (() | (Lower_Arc (rectangle (a,b,c,d)))) ; :: thesis: verum
end;
then reconsider f3 = f2 as Function of I[01],(() | (Lower_Arc (rectangle (a,b,c,d)))) by ;
A28: 0 in [.0,1.] by XXREAL_1:1;
0 in [.0,(1 / 2).] by XXREAL_1:1;
then A29: f3 . 0 = ((1 - (2 * 0)) * |[b,d]|) + ((2 * 0) * |[b,c]|) by
.= (1 * |[b,d]|) + (0. ()) by RLVECT_1:10
.= |[b,d]| + (0. ()) by RLVECT_1:def 8
.= |[b,d]| by RLVECT_1:4
.= E-max (rectangle (a,b,c,d)) by A1, A2, Th46 ;
A30: 1 in [.0,1.] by XXREAL_1:1;
1 in [.(1 / 2),1.] by XXREAL_1:1;
then A31: f3 . 1 = ((1 - ((2 * 1) - 1)) * |[b,c]|) + (((2 * 1) - 1) * |[a,c]|) by
.= (0 * |[b,c]|) + |[a,c]| by RLVECT_1:def 8
.= (0. ()) + |[a,c]| by RLVECT_1:10
.= |[a,c]| by RLVECT_1:4
.= W-min (rectangle (a,b,c,d)) by A1, A2, Th46 ;
A32: for r being Real st r in [.0,(1 / 2).] holds
f3 . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|)
proof
let r be Real; :: thesis: ( r in [.0,(1 / 2).] implies f3 . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) )
assume A33: r in [.0,(1 / 2).] ; :: thesis: f3 . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|)
then A34: 0 <= r by XXREAL_1:1;
r <= 1 / 2 by ;
then r <= 1 by XXREAL_0:2;
then r in [.0,1.] by ;
hence f3 . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) by ; :: thesis: verum
end;
A35: for r being Real st r in [.(1 / 2),1.] holds
f3 . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|)
proof
let r be Real; :: thesis: ( r in [.(1 / 2),1.] implies f3 . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) )
assume A36: r in [.(1 / 2),1.] ; :: thesis: f3 . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|)
then A37: 1 / 2 <= r by XXREAL_1:1;
r <= 1 by ;
then r in [.0,1.] by ;
hence f3 . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) by ; :: thesis: verum
end;
A38: for p being Point of () st p in LSeg (|[b,d]|,|[b,c]|) holds
( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f3 . ((((p `2) - d) / (c - d)) / 2) = p )
proof
let p be Point of (); :: thesis: ( p in LSeg (|[b,d]|,|[b,c]|) implies ( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f3 . ((((p `2) - d) / (c - d)) / 2) = p ) )
assume A39: p in LSeg (|[b,d]|,|[b,c]|) ; :: thesis: ( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f3 . ((((p `2) - d) / (c - d)) / 2) = p )
A40: |[b,d]| `2 = d by EUCLID:52;
A41: |[b,c]| `2 = c by EUCLID:52;
then A42: c <= p `2 by ;
A43: p `2 <= d by ;
d - c > 0 by ;
then A44: - (d - c) < - 0 by XREAL_1:24;
d - (p `2) >= 0 by ;
then A45: - (d - (p `2)) <= - 0 ;
(p `2) - d >= c - d by ;
then ((p `2) - d) / (c - d) <= (c - d) / (c - d) by ;
then ((p `2) - d) / (c - d) <= 1 by ;
then A46: (((p `2) - d) / (c - d)) / 2 <= 1 / 2 by XREAL_1:72;
set r = (((p `2) - d) / (c - d)) / 2;
(((p `2) - d) / (c - d)) / 2 in [.0,(1 / 2).] by ;
then f3 . ((((p `2) - d) / (c - d)) / 2) = ((1 - (2 * ((((p `2) - d) / (c - d)) / 2))) * |[b,d]|) + ((2 * ((((p `2) - d) / (c - d)) / 2)) * |[b,c]|) by A32
.= |[((1 - (2 * ((((p `2) - d) / (c - d)) / 2))) * b),((1 - (2 * ((((p `2) - d) / (c - d)) / 2))) * d)]| + ((2 * ((((p `2) - d) / (c - d)) / 2)) * |[b,c]|) by EUCLID:58
.= |[((1 - (2 * ((((p `2) - d) / (c - d)) / 2))) * b),((1 - (2 * ((((p `2) - d) / (c - d)) / 2))) * d)]| + |[((2 * ((((p `2) - d) / (c - d)) / 2)) * b),((2 * ((((p `2) - d) / (c - d)) / 2)) * c)]| by EUCLID:58
.= |[(((1 * b) - ((2 * ((((p `2) - d) / (c - d)) / 2)) * b)) + ((2 * ((((p `2) - d) / (c - d)) / 2)) * b)),(((1 - (2 * ((((p `2) - d) / (c - d)) / 2))) * d) + ((2 * ((((p `2) - d) / (c - d)) / 2)) * c))]| by EUCLID:56
.= |[b,((1 * d) + ((((p `2) - d) / (c - d)) * (c - d)))]|
.= |[b,((1 * d) + ((p `2) - d))]| by
.= |[(p `1),(p `2)]| by
.= p by EUCLID:53 ;
hence ( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f3 . ((((p `2) - d) / (c - d)) / 2) = p ) by ; :: thesis: verum
end;
A47: for p being Point of () st p in LSeg (|[b,c]|,|[a,c]|) holds
( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f3 . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p )
proof
let p be Point of (); :: thesis: ( p in LSeg (|[b,c]|,|[a,c]|) implies ( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f3 . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) )
assume A48: p in LSeg (|[b,c]|,|[a,c]|) ; :: thesis: ( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f3 . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p )
A49: |[b,c]| `1 = b by EUCLID:52;
A50: |[a,c]| `1 = a by EUCLID:52;
then A51: a <= p `1 by ;
A52: p `1 <= b by ;
b - a > 0 by ;
then A53: - (b - a) < - 0 by XREAL_1:24;
b - (p `1) >= 0 by ;
then A54: - (b - (p `1)) <= - 0 ;
then A55: ((((p `1) - b) / (a - b)) / 2) + (1 / 2) >= 0 + (1 / 2) by ;
(p `1) - b >= a - b by ;
then ((p `1) - b) / (a - b) <= (a - b) / (a - b) by ;
then ((p `1) - b) / (a - b) <= 1 by ;
then (((p `1) - b) / (a - b)) / 2 <= 1 / 2 by XREAL_1:72;
then A56: ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= (1 / 2) + (1 / 2) by XREAL_1:7;
set r = ((((p `1) - b) / (a - b)) / 2) + (1 / 2);
((((p `1) - b) / (a - b)) / 2) + (1 / 2) in [.(1 / 2),1.] by ;
then f3 . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = ((1 - ((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * |[b,c]|) + (((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1) * |[a,c]|) by A35
.= |[((1 - ((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * b),((1 - ((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * c)]| + (((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1) * |[a,c]|) by EUCLID:58
.= |[((1 - ((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * b),((1 - ((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * c)]| + |[(((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1) * a),(((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1) * c)]| by EUCLID:58
.= |[(((1 - ((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * b) + (((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1) * a)),(((1 * c) - (((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1) * c)) + (((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1) * c))]| by EUCLID:56
.= |[((1 * b) + ((((p `1) - b) / (a - b)) * (a - b))),c]|
.= |[((1 * b) + ((p `1) - b)),c]| by
.= |[(p `1),(p `2)]| by
.= p by EUCLID:53 ;
hence ( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f3 . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) by ; :: thesis: verum
end;
reconsider B00 = [.0,1.] as Subset of R^1 by TOPMETR:17;
reconsider B01 = B00 as non empty Subset of R^1 by XXREAL_1:1;
I[01] = R^1 | B01 by ;
then consider h1 being Function of I[01],R^1 such that
A57: for p being Point of I[01] holds h1 . p = p and
A58: h1 is continuous by Th6;
consider h2 being Function of I[01],R^1 such that
A59: for p being Point of I[01]
for r1 being Real st h1 . p = r1 holds
h2 . p = 2 * r1 and
A60: h2 is continuous by ;
consider h5 being Function of I[01],R^1 such that
A61: for p being Point of I[01]
for r1 being Real st h2 . p = r1 holds
h5 . p = 1 - r1 and
A62: h5 is continuous by ;
consider h3 being Function of I[01],R^1 such that
A63: for p being Point of I[01]
for r1 being Real st h2 . p = r1 holds
h3 . p = r1 - 1 and
A64: h3 is continuous by ;
consider h4 being Function of I[01],R^1 such that
A65: for p being Point of I[01]
for r1 being Real st h3 . p = r1 holds
h4 . p = 1 - r1 and
A66: h4 is continuous by ;
consider g1 being Function of I[01],() such that
A67: for r being Point of I[01] holds g1 . r = ((h5 . r) * |[b,d]|) + ((h2 . r) * |[b,c]|) and
A68: g1 is continuous by ;
A69: for r being Point of I[01]
for s being Real st r = s holds
g1 . r = ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|)
proof
let r be Point of I[01]; :: thesis: for s being Real st r = s holds
g1 . r = ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|)

let s be Real; :: thesis: ( r = s implies g1 . r = ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|) )
assume A70: r = s ; :: thesis: g1 . r = ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|)
g1 . r = ((h5 . r) * |[b,d]|) + ((h2 . r) * |[b,c]|) by A67
.= ((1 - (2 * (h1 . r))) * |[b,d]|) + ((h2 . r) * |[b,c]|) by
.= ((1 - (2 * (h1 . r))) * |[b,d]|) + ((2 * (h1 . r)) * |[b,c]|) by A59
.= ((1 - (2 * s)) * |[b,d]|) + ((2 * (h1 . r)) * |[b,c]|) by
.= ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|) by ;
hence g1 . r = ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|) ; :: thesis: verum
end;
consider g2 being Function of I[01],() such that
A71: for r being Point of I[01] holds g2 . r = ((h4 . r) * |[b,c]|) + ((h3 . r) * |[a,c]|) and
A72: g2 is continuous by ;
A73: for r being Point of I[01]
for s being Real st r = s holds
g2 . r = ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|)
proof
let r be Point of I[01]; :: thesis: for s being Real st r = s holds
g2 . r = ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|)

let s be Real; :: thesis: ( r = s implies g2 . r = ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|) )
assume A74: r = s ; :: thesis: g2 . r = ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|)
g2 . r = ((h4 . r) * |[b,c]|) + ((h3 . r) * |[a,c]|) by A71
.= ((1 - ((h2 . r) - 1)) * |[b,c]|) + ((h3 . r) * |[a,c]|) by
.= ((1 - ((h2 . r) - 1)) * |[b,c]|) + (((h2 . r) - 1) * |[a,c]|) by A63
.= ((1 - ((2 * (h1 . r)) - 1)) * |[b,c]|) + (((h2 . r) - 1) * |[a,c]|) by A59
.= ((1 - ((2 * (h1 . r)) - 1)) * |[b,c]|) + (((2 * (h1 . r)) - 1) * |[a,c]|) by A59
.= ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * (h1 . r)) - 1) * |[a,c]|) by
.= ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|) by ;
hence g2 . r = ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|) ; :: thesis: verum
end;
reconsider B11 = [.0,(1 / 2).] as non empty Subset of I[01] by ;
A75: dom (g1 | B11) = (dom g1) /\ B11 by RELAT_1:61
.= the carrier of I[01] /\ B11 by FUNCT_2:def 1
.= B11 by XBOOLE_1:28
.= the carrier of (I[01] | B11) by PRE_TOPC:8 ;
rng (g1 | B11) c= the carrier of () ;
then reconsider g11 = g1 | B11 as Function of (I[01] | B11),() by ;
A76: TOP-REAL 2 is SubSpace of TOP-REAL 2 by TSEP_1:2;
then A77: g11 is continuous by ;
reconsider B22 = [.(1 / 2),1.] as non empty Subset of I[01] by ;
A78: dom (g2 | B22) = (dom g2) /\ B22 by RELAT_1:61
.= the carrier of I[01] /\ B22 by FUNCT_2:def 1
.= B22 by XBOOLE_1:28
.= the carrier of (I[01] | B22) by PRE_TOPC:8 ;
rng (g2 | B22) c= the carrier of () ;
then reconsider g22 = g2 | B22 as Function of (I[01] | B22),() by ;
A79: g22 is continuous by ;
A80: B11 = [#] (I[01] | B11) by PRE_TOPC:def 5;
A81: B22 = [#] (I[01] | B22) by PRE_TOPC:def 5;
A82: B11 is closed by Th4;
A83: B22 is closed by Th4;
A84: ([#] (I[01] | B11)) \/ ([#] (I[01] | B22)) = [#] I[01] by ;
for p being object st p in ([#] (I[01] | B11)) /\ ([#] (I[01] | B22)) holds
g11 . p = g22 . p
proof
let p be object ; :: thesis: ( p in ([#] (I[01] | B11)) /\ ([#] (I[01] | B22)) implies g11 . p = g22 . p )
assume A85: p in ([#] (I[01] | B11)) /\ ([#] (I[01] | B22)) ; :: thesis: g11 . p = g22 . p
then A86: p in [#] (I[01] | B11) by XBOOLE_0:def 4;
A87: p in [#] (I[01] | B22) by A85;
A88: p in B11 by ;
A89: p in B22 by ;
reconsider rp = p as Real by A88;
A90: rp <= 1 / 2 by ;
rp >= 1 / 2 by ;
then rp = 1 / 2 by ;
then A91: 2 * rp = 1 ;
thus g11 . p = g1 . p by
.= ((1 - 1) * |[b,d]|) + (1 * |[b,c]|) by
.= (0. ()) + (1 * |[b,c]|) by RLVECT_1:10
.= ((1 - 0) * |[b,c]|) + ((1 - 1) * |[a,c]|) by RLVECT_1:10
.= g2 . p by
.= g22 . p by ; :: thesis: verum
end;
then consider h being Function of I[01],() such that
A92: h = g11 +* g22 and
A93: h is continuous by ;
A94: dom f3 = dom h by Th5;
A95: dom f3 = the carrier of I[01] by Th5;
for x being object st x in dom f2 holds
f3 . x = h . x
proof
let x be object ; :: thesis: ( x in dom f2 implies f3 . x = h . x )
assume A96: x in dom f2 ; :: thesis: f3 . x = h . x
then reconsider rx = x as Real by A95;
A97: 0 <= rx by ;
A98: rx <= 1 by ;
per cases ( rx < 1 / 2 or rx >= 1 / 2 ) ;
suppose A99: rx < 1 / 2 ; :: thesis: f3 . x = h . x
then A100: rx in [.0,(1 / 2).] by ;
not rx in dom g22 by ;
then h . rx = g11 . rx by
.= g1 . rx by
.= ((1 - (2 * rx)) * |[b,d]|) + ((2 * rx) * |[b,c]|) by
.= f3 . rx by ;
hence f3 . x = h . x ; :: thesis: verum
end;
suppose rx >= 1 / 2 ; :: thesis: f3 . x = h . x
then A101: rx in [.(1 / 2),1.] by ;
then rx in [#] (I[01] | B22) by PRE_TOPC:def 5;
then h . rx = g22 . rx by
.= g2 . rx by
.= ((1 - ((2 * rx) - 1)) * |[b,c]|) + (((2 * rx) - 1) * |[a,c]|) by
.= f3 . rx by ;
hence f3 . x = h . x ; :: thesis: verum
end;
end;
end;
then A102: f2 = h by ;
for x1, x2 being object st x1 in dom f3 & x2 in dom f3 & f3 . x1 = f3 . x2 holds
x1 = x2
proof
let x1, x2 be object ; :: thesis: ( x1 in dom f3 & x2 in dom f3 & f3 . x1 = f3 . x2 implies x1 = x2 )
assume that
A103: x1 in dom f3 and
A104: x2 in dom f3 and
A105: f3 . x1 = f3 . x2 ; :: thesis: x1 = x2
reconsider r1 = x1 as Real by A103;
reconsider r2 = x2 as Real by A104;
A106: (LSeg (|[b,d]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[a,c]|)) = {|[b,c]|} by A1, A2, Th32;
now :: thesis: ( ( x1 in [.0,(1 / 2).] & x2 in [.0,(1 / 2).] & x1 = x2 ) or ( x1 in [.0,(1 / 2).] & x2 in [.(1 / 2),1.] & x1 = x2 ) or ( x1 in [.(1 / 2),1.] & x2 in [.0,(1 / 2).] & x1 = x2 ) or ( x1 in [.(1 / 2),1.] & x2 in [.(1 / 2),1.] & x1 = x2 ) )
per cases ( ( x1 in [.0,(1 / 2).] & x2 in [.0,(1 / 2).] ) or ( x1 in [.0,(1 / 2).] & x2 in [.(1 / 2),1.] ) or ( x1 in [.(1 / 2),1.] & x2 in [.0,(1 / 2).] ) or ( x1 in [.(1 / 2),1.] & x2 in [.(1 / 2),1.] ) ) by ;
case A107: ( x1 in [.0,(1 / 2).] & x2 in [.0,(1 / 2).] ) ; :: thesis: x1 = x2
then f3 . r1 = ((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|) by A32;
then ((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|) = ((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|) by ;
then (((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|)) - ((2 * r1) * |[b,c]|) = (1 - (2 * r1)) * |[b,d]| by RLVECT_4:1;
then ((1 - (2 * r2)) * |[b,d]|) + (((2 * r2) * |[b,c]|) - ((2 * r1) * |[b,c]|)) = (1 - (2 * r1)) * |[b,d]| by RLVECT_1:def 3;
then ((1 - (2 * r2)) * |[b,d]|) + (((2 * r2) - (2 * r1)) * |[b,c]|) = (1 - (2 * r1)) * |[b,d]| by RLVECT_1:35;
then (((2 * r2) - (2 * r1)) * |[b,c]|) + (((1 - (2 * r2)) * |[b,d]|) - ((1 - (2 * r1)) * |[b,d]|)) = ((1 - (2 * r1)) * |[b,d]|) - ((1 - (2 * r1)) * |[b,d]|) by RLVECT_1:def 3;
then (((2 * r2) - (2 * r1)) * |[b,c]|) + (((1 - (2 * r2)) * |[b,d]|) - ((1 - (2 * r1)) * |[b,d]|)) = 0. () by RLVECT_1:5;
then (((2 * r2) - (2 * r1)) * |[b,c]|) + (((1 - (2 * r2)) - (1 - (2 * r1))) * |[b,d]|) = 0. () by RLVECT_1:35;
then (((2 * r2) - (2 * r1)) * |[b,c]|) + ((- ((2 * r2) - (2 * r1))) * |[b,d]|) = 0. () ;
then (((2 * r2) - (2 * r1)) * |[b,c]|) + (- (((2 * r2) - (2 * r1)) * |[b,d]|)) = 0. () by RLVECT_1:79;
then (((2 * r2) - (2 * r1)) * |[b,c]|) - (((2 * r2) - (2 * r1)) * |[b,d]|) = 0. () ;
then ((2 * r2) - (2 * r1)) * (|[b,c]| - |[b,d]|) = 0. () by RLVECT_1:34;
then ( (2 * r2) - (2 * r1) = 0 or |[b,c]| - |[b,d]| = 0. () ) by RLVECT_1:11;
then ( (2 * r2) - (2 * r1) = 0 or |[b,c]| = |[b,d]| ) by RLVECT_1:21;
then ( (2 * r2) - (2 * r1) = 0 or d = |[b,c]| `2 ) by EUCLID:52;
hence x1 = x2 by ; :: thesis: verum
end;
case A108: ( x1 in [.0,(1 / 2).] & x2 in [.(1 / 2),1.] ) ; :: thesis: x1 = x2
then A109: f3 . r1 = ((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|) by A32;
A110: 0 <= r1 by ;
r1 <= 1 / 2 by ;
then r1 * 2 <= (1 / 2) * 2 by XREAL_1:64;
then A111: f3 . r1 in LSeg (|[b,d]|,|[b,c]|) by ;
A112: f3 . r2 = ((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|) by ;
A113: 1 / 2 <= r2 by ;
A114: r2 <= 1 by ;
r2 * 2 >= (1 / 2) * 2 by ;
then A115: (2 * r2) - 1 >= 0 by XREAL_1:48;
2 * 1 >= 2 * r2 by ;
then (1 + 1) - 1 >= (2 * r2) - 1 by XREAL_1:9;
then f3 . r2 in { (((1 - lambda) * |[b,c]|) + (lambda * |[a,c]|)) where lambda is Real : ( 0 <= lambda & lambda <= 1 ) } by ;
then f3 . r1 in (LSeg (|[b,d]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[a,c]|)) by ;
then A116: f3 . r1 = |[b,c]| by ;
then (((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|)) + (- |[b,c]|) = 0. () by ;
then (((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|)) + ((- 1) * |[b,c]|) = 0. () by RLVECT_1:16;
then ((1 - (2 * r1)) * |[b,d]|) + (((2 * r1) * |[b,c]|) + ((- 1) * |[b,c]|)) = 0. () by RLVECT_1:def 3;
then ((1 - (2 * r1)) * |[b,d]|) + (((2 * r1) + (- 1)) * |[b,c]|) = 0. () by RLVECT_1:def 6;
then ((1 - (2 * r1)) * |[b,d]|) + ((- (1 - (2 * r1))) * |[b,c]|) = 0. () ;
then ((1 - (2 * r1)) * |[b,d]|) + (- ((1 - (2 * r1)) * |[b,c]|)) = 0. () by RLVECT_1:79;
then ((1 - (2 * r1)) * |[b,d]|) - ((1 - (2 * r1)) * |[b,c]|) = 0. () ;
then (1 - (2 * r1)) * (|[b,d]| - |[b,c]|) = 0. () by RLVECT_1:34;
then ( 1 - (2 * r1) = 0 or |[b,d]| - |[b,c]| = 0. () ) by RLVECT_1:11;
then ( 1 - (2 * r1) = 0 or |[b,d]| = |[b,c]| ) by RLVECT_1:21;
then A117: ( 1 - (2 * r1) = 0 or d = |[b,c]| `2 ) by EUCLID:52;
(((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|)) + (- |[b,c]|) = 0. () by ;
then (((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|)) + ((- 1) * |[b,c]|) = 0. () by RLVECT_1:16;
then (((2 * r2) - 1) * |[a,c]|) + (((1 - ((2 * r2) - 1)) * |[b,c]|) + ((- 1) * |[b,c]|)) = 0. () by RLVECT_1:def 3;
then (((2 * r2) - 1) * |[a,c]|) + (((1 - ((2 * r2) - 1)) + (- 1)) * |[b,c]|) = 0. () by RLVECT_1:def 6;
then (((2 * r2) - 1) * |[a,c]|) + ((- ((2 * r2) - 1)) * |[b,c]|) = 0. () ;
then (((2 * r2) - 1) * |[a,c]|) + (- (((2 * r2) - 1) * |[b,c]|)) = 0. () by RLVECT_1:79;
then (((2 * r2) - 1) * |[a,c]|) - (((2 * r2) - 1) * |[b,c]|) = 0. () ;
then ((2 * r2) - 1) * (|[a,c]| - |[b,c]|) = 0. () by RLVECT_1:34;
then ( (2 * r2) - 1 = 0 or |[a,c]| - |[b,c]| = 0. () ) by RLVECT_1:11;
then ( (2 * r2) - 1 = 0 or |[a,c]| = |[b,c]| ) by RLVECT_1:21;
then ( (2 * r2) - 1 = 0 or a = |[b,c]| `1 ) by EUCLID:52;
hence x1 = x2 by ; :: thesis: verum
end;
case A118: ( x1 in [.(1 / 2),1.] & x2 in [.0,(1 / 2).] ) ; :: thesis: x1 = x2
then A119: f3 . r2 = ((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|) by A32;
A120: 0 <= r2 by ;
r2 <= 1 / 2 by ;
then r2 * 2 <= (1 / 2) * 2 by XREAL_1:64;
then A121: f3 . r2 in LSeg (|[b,d]|,|[b,c]|) by ;
A122: f3 . r1 = ((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|) by ;
A123: 1 / 2 <= r1 by ;
A124: r1 <= 1 by ;
r1 * 2 >= (1 / 2) * 2 by ;
then A125: (2 * r1) - 1 >= 0 by XREAL_1:48;
2 * 1 >= 2 * r1 by ;
then (1 + 1) - 1 >= (2 * r1) - 1 by XREAL_1:9;
then f3 . r1 in { (((1 - lambda) * |[b,c]|) + (lambda * |[a,c]|)) where lambda is Real : ( 0 <= lambda & lambda <= 1 ) } by ;
then f3 . r2 in (LSeg (|[b,d]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[a,c]|)) by ;
then A126: f3 . r2 = |[b,c]| by ;
then (((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|)) + (- |[b,c]|) = 0. () by ;
then (((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|)) + ((- 1) * |[b,c]|) = 0. () by RLVECT_1:16;
then ((1 - (2 * r2)) * |[b,d]|) + (((2 * r2) * |[b,c]|) + ((- 1) * |[b,c]|)) = 0. () by RLVECT_1:def 3;
then ((1 - (2 * r2)) * |[b,d]|) + (((2 * r2) + (- 1)) * |[b,c]|) = 0. () by RLVECT_1:def 6;
then ((1 - (2 * r2)) * |[b,d]|) + ((- (1 - (2 * r2))) * |[b,c]|) = 0. () ;
then ((1 - (2 * r2)) * |[b,d]|) + (- ((1 - (2 * r2)) * |[b,c]|)) = 0. () by RLVECT_1:79;
then ((1 - (2 * r2)) * |[b,d]|) - ((1 - (2 * r2)) * |[b,c]|) = 0. () ;
then (1 - (2 * r2)) * (|[b,d]| - |[b,c]|) = 0. () by RLVECT_1:34;
then ( 1 - (2 * r2) = 0 or |[b,d]| - |[b,c]| = 0. () ) by RLVECT_1:11;
then ( 1 - (2 * r2) = 0 or |[b,d]| = |[b,c]| ) by RLVECT_1:21;
then A127: ( 1 - (2 * r2) = 0 or d = |[b,c]| `2 ) by EUCLID:52;
(((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|)) + (- |[b,c]|) = 0. () by ;
then (((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|)) + ((- 1) * |[b,c]|) = 0. () by RLVECT_1:16;
then (((2 * r1) - 1) * |[a,c]|) + (((1 - ((2 * r1) - 1)) * |[b,c]|) + ((- 1) * |[b,c]|)) = 0. () by RLVECT_1:def 3;
then (((2 * r1) - 1) * |[a,c]|) + (((1 - ((2 * r1) - 1)) + (- 1)) * |[b,c]|) = 0. () by RLVECT_1:def 6;
then (((2 * r1) - 1) * |[a,c]|) + ((- ((2 * r1) - 1)) * |[b,c]|) = 0. () ;
then (((2 * r1) - 1) * |[a,c]|) + (- (((2 * r1) - 1) * |[b,c]|)) = 0. () by RLVECT_1:79;
then (((2 * r1) - 1) * |[a,c]|) - (((2 * r1) - 1) * |[b,c]|) = 0. () ;
then ((2 * r1) - 1) * (|[a,c]| - |[b,c]|) = 0. () by RLVECT_1:34;
then ( (2 * r1) - 1 = 0 or |[a,c]| - |[b,c]| = 0. () ) by RLVECT_1:11;
then ( (2 * r1) - 1 = 0 or |[a,c]| = |[b,c]| ) by RLVECT_1:21;
then ( (2 * r1) - 1 = 0 or a = |[b,c]| `1 ) by EUCLID:52;
hence x1 = x2 by ; :: thesis: verum
end;
case A128: ( x1 in [.(1 / 2),1.] & x2 in [.(1 / 2),1.] ) ; :: thesis: x1 = x2
then f3 . r1 = ((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|) by A35;
then ((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|) = ((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|) by ;
then (((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|)) - (((2 * r1) - 1) * |[a,c]|) = (1 - ((2 * r1) - 1)) * |[b,c]| by RLVECT_4:1;
then ((1 - ((2 * r2) - 1)) * |[b,c]|) + ((((2 * r2) - 1) * |[a,c]|) - (((2 * r1) - 1) * |[a,c]|)) = (1 - ((2 * r1) - 1)) * |[b,c]| by RLVECT_1:def 3;
then ((1 - ((2 * r2) - 1)) * |[b,c]|) + ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) = (1 - ((2 * r1) - 1)) * |[b,c]| by RLVECT_1:35;
then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + (((1 - ((2 * r2) - 1)) * |[b,c]|) - ((1 - ((2 * r1) - 1)) * |[b,c]|)) = ((1 - ((2 * r1) - 1)) * |[b,c]|) - ((1 - ((2 * r1) - 1)) * |[b,c]|) by RLVECT_1:def 3;
then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + (((1 - ((2 * r2) - 1)) * |[b,c]|) - ((1 - ((2 * r1) - 1)) * |[b,c]|)) = 0. () by RLVECT_1:5;
then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + (((1 - ((2 * r2) - 1)) - (1 - ((2 * r1) - 1))) * |[b,c]|) = 0. () by RLVECT_1:35;
then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + ((- (((2 * r2) - 1) - ((2 * r1) - 1))) * |[b,c]|) = 0. () ;
then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + (- ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b,c]|)) = 0. () by RLVECT_1:79;
then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) - ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b,c]|) = 0. () ;
then (((2 * r2) - 1) - ((2 * r1) - 1)) * (|[a,c]| - |[b,c]|) = 0. () by RLVECT_1:34;
then ( ((2 * r2) - 1) - ((2 * r1) - 1) = 0 or |[a,c]| - |[b,c]| = 0. () ) by RLVECT_1:11;
then ( ((2 * r2) - 1) - ((2 * r1) - 1) = 0 or |[a,c]| = |[b,c]| ) by RLVECT_1:21;
then ( ((2 * r2) - 1) - ((2 * r1) - 1) = 0 or a = |[b,c]| `1 ) by EUCLID:52;
hence x1 = x2 by ; :: thesis: verum
end;
end;
end;
hence x1 = x2 ; :: thesis: verum
end;
then A129: f3 is one-to-one by FUNCT_1:def 4;
[#] (() | (Lower_Arc (rectangle (a,b,c,d)))) c= rng f3
proof
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in [#] (() | (Lower_Arc (rectangle (a,b,c,d)))) or y in rng f3 )
assume y in [#] (() | (Lower_Arc (rectangle (a,b,c,d)))) ; :: thesis: y in rng f3
then A130: y in Lower_Arc (rectangle (a,b,c,d)) by PRE_TOPC:def 5;
then reconsider q = y as Point of () ;
A131: Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52;
now :: thesis: ( ( q in LSeg (|[b,d]|,|[b,c]|) & y in rng f3 ) or ( q in LSeg (|[b,c]|,|[a,c]|) & y in rng f3 ) )
per cases ( q in LSeg (|[b,d]|,|[b,c]|) or q in LSeg (|[b,c]|,|[a,c]|) ) by ;
case A132: q in LSeg (|[b,d]|,|[b,c]|) ; :: thesis: y in rng f3
then A133: 0 <= (((q `2) - d) / (c - d)) / 2 by A38;
A134: (((q `2) - d) / (c - d)) / 2 <= 1 by ;
A135: f3 . ((((q `2) - d) / (c - d)) / 2) = q by ;
(((q `2) - d) / (c - d)) / 2 in [.0,1.] by ;
hence y in rng f3 by ; :: thesis: verum
end;
case A136: q in LSeg (|[b,c]|,|[a,c]|) ; :: thesis: y in rng f3
then A137: 0 <= ((((q `1) - b) / (a - b)) / 2) + (1 / 2) by A47;
A138: ((((q `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 by ;
A139: f3 . (((((q `1) - b) / (a - b)) / 2) + (1 / 2)) = q by ;
((((q `1) - b) / (a - b)) / 2) + (1 / 2) in [.0,1.] by ;
hence y in rng f3 by ; :: thesis: verum
end;
end;
end;
hence y in rng f3 ; :: thesis: verum
end;
then A140: rng f3 = [#] (() | (Lower_Arc (rectangle (a,b,c,d)))) ;
I[01] is compact by ;
then A141: f3 is being_homeomorphism by ;
rng f3 = Lower_Arc (rectangle (a,b,c,d)) by ;
hence ex f being Function of I[01],(() | (Lower_Arc (rectangle (a,b,c,d)))) st
( f is being_homeomorphism & f . 0 = E-max (rectangle (a,b,c,d)) & f . 1 = W-min (rectangle (a,b,c,d)) & rng f = Lower_Arc (rectangle (a,b,c,d)) & ( for r being Real st r in [.0,(1 / 2).] holds
f . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) ) & ( for r being Real st r in [.(1 / 2),1.] holds
f . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) ) & ( for p being Point of () st p in LSeg (|[b,d]|,|[b,c]|) holds
( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f . ((((p `2) - d) / (c - d)) / 2) = p ) ) & ( for p being Point of () st p in LSeg (|[b,c]|,|[a,c]|) holds
( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) ) ) by A29, A31, A32, A35, A38, A47, A141; :: thesis: verum